BINOMIAL THEOREM - 4 (For Positive Integral Index - Problem Solving )
Summation of Series (involving binomial coefficients)
1. Bino-geometric series
${ }^{n} \mathrm{C} _{0}+{ }^{n} \mathrm{C} _{1} \mathrm{x}+{ }^{\mathrm{n}} \mathrm{C} _{2} \mathrm{x}^{2}$. $+{ }^{n} C _{n} x^{n}=(1+x)^{n}$
2. Bino-arithmetic series
$a^{n} C _{o}+(a+d)^{n} C _{1}+(a+2 d)^{n} C _{2}+\ldots \ldots \ldots \ldots \ldots \ldots . .+(a+n d)^{n} C _{n}$
This series is the sum of the products of corresponding terms of
${ }^{\mathrm{n}} \mathrm{C} _{0},{ }^{\mathrm{n}} \mathrm{C} _{1},{ }^{\mathrm{n}} \mathrm{C} _{2}, \ldots \ldots \ldots \ldots . . .{ }^{\mathrm{n}} \mathrm{C} _{\mathrm{n}}$ (binomial coefficients) and $\mathrm{a}, \mathrm{a}+\mathrm{d}, \mathrm{a}+2 \mathrm{~d}$, $……….,\mathrm{a}+\mathrm{nd}$ (arithmetic progression)
Such series can be solved either by
(i). eliminating $\mathrm{r}$ in the multiplier of binomial coefficient from the $(\mathrm{r}+1)^{\text {th }}$ terms of the series (i.e. using $r^{n} C _{r}=n^{n-1} C _{r-1}$ )
or
(ii). Differentiating the expansion of $\mathrm{x}^{\mathrm{a}}\left(1+\mathrm{x}^{\mathrm{d}}\right)^{\mathrm{n}}$ or (If product of two or more numericals occur, then differentiate again and again till we get the desired result)
3. Bino-harmonic series
$\frac{{ }^{n} C _{0}}{a}+\frac{{ }^{n} C _{1}}{a+d}+\frac{{ }^{n} C _{2}}{a+2 d}+\ldots \ldots . .+\frac{{ }^{n} C _{n}}{a+n d}$
This series is the sum of the products of corresponding terms of
${ }^{\mathrm{n}} \mathrm{C} _{0},{ }^{\mathrm{n}} \mathrm{C} _{1},{ }^{\mathrm{n}} \mathrm{C} _{2}, \ldots \ldots \ldots \ldots . .{ }^{\mathrm{n}} \mathrm{C} _{\mathrm{n}}$ (binomial coefficients) and
$\frac{1}{a}, \frac{1}{a+d}, \frac{1}{a+2 d}, \ldots \ldots \ldots \ldots \ldots . . . \frac{1}{a+n d}$ (harmonic progression)
Such seris can be solved either by
(i). eliminating $\mathrm{r}$ in the multiplier of binomial coefficient from the $(\mathrm{r}+1)^{\text {th }}$ term of the series
(ie using $\frac{1}{\mathrm{r}+1}{ }^{\mathrm{n}} \mathrm{C} _{\mathrm{r}}=\frac{1}{\mathrm{n}+1}{ }^{\mathrm{n}+1} \mathrm{C} _{\mathrm{r}+1}$ ) or
(ii). integrating suitable expansion
Note
(i). If the sum contains $\mathrm{C} _{0}, \mathrm{C} _{1}, \mathrm{C} _{2} \ldots \ldots \ldots \ldots \mathrm{C} _{\mathrm{n}}$ are all positive signs, integrate between limits $0$ to $1$
(ii). If the sum contains alternate signs (i.e. $&-$ ) then integrate between limits $-1$ to $0$
(iii). If the sum contains odd coefficients (i.e. $\mathrm{C} _{0}, \mathrm{C} _{2}, \mathrm{C} _{4}, \ldots \ldots .$. ) then integrate between $-1$ to $+1$ .
(iv). If the sum contains even coefficient (i.e. $\mathrm{C} _{1}, \mathrm{C} _{3}, \mathrm{C} _{5}, \ldots \ldots .$. ) then find the difference between (i) & (iii) and then divide by $2$
(v). If in denominator of binomial coefficient is product of two numericals then integrate two times first time take limits between 0 to $x$ and second time take suitable limits
4. Bino-binomial series.
${ }^{\mathrm{n}} \mathrm{C} _{0}{ }^{\mathrm{n}} \mathrm{C} _{\mathrm{r}}+{ }^{\mathrm{n}} \mathrm{C} _{1}{ }^{\mathrm{n}} \mathrm{C} _{\mathrm{r}+1}+{ }^{\mathrm{n}} \mathrm{C} _{2}{ }^{\mathrm{n}} \mathrm{C} _{\mathrm{r}+2}+\ldots \ldots \ldots .+{ }^{\mathrm{n}} \mathrm{C} _{\mathrm{n}-\mathrm{r}}{ }^{\mathrm{n}} \mathrm{C} _{\mathrm{n}} \quad$ or
${ }^{\mathrm{m}} \mathrm{C} _{0}{ }^{\mathrm{n}} \mathrm{C} _{\mathrm{r}}+{ }^{\mathrm{m}} \mathrm{C} _{1}{ }^{\mathrm{n}} \mathrm{C} _{\mathrm{r}-1}+{ }^{\mathrm{m}} \mathrm{C} _{2}{ }^{\mathrm{n}} \mathrm{C} _{\mathrm{r}-2}+\ldots \ldots \ldots . .+{ }^{\mathrm{m}} \mathrm{C} _{\mathrm{r}}{ }^{\mathrm{n}} \mathrm{C} _{0}$
Such series can be solved by multiplying two expansions, one involving the first factors as coefficient and the other involving the second factors as coefficients and finally equating coefficients of a suitable power of $x$ on both sides.
Binomial coefficients
1. $\mathrm{C}_0+\mathrm{C}_1+\mathrm{C}_2+\mathrm{C}_3+\ldots \ldots \ldots \ldots .2^{\mathrm{n}} $
2. $\mathrm{C}_0-\mathrm{C}_1+\mathrm{C}_2-\mathrm{C}_3+\ldots \ldots \ldots \ldots .=0$
3. $ C _0- C _1+ C _2- C _3+\ldots \ldots \ldots \ldots+ C _ r (-1)^ r = ^ n -1 C _ r (-1)^ r ; r < n $
4. $C _ o + C _2+ C _4+ C _6+\ldots \ldots \ldots \ldots . .=2^ n -1$
5. $\mathrm{C}_1+\mathrm{C}_3+\mathrm{C}_5+\mathrm{C}_7+\ldots \ldots \ldots \ldots=2^{\mathrm{n}-1} $
6. $\mathrm{C}_0-\mathrm{C}_2+\mathrm{C}_4-\mathrm{C}_6+\ldots \ldots \ldots=(\sqrt{2})^{\mathrm{n}} \cos \frac{\mathrm{n} \pi}{4} $
7. $\mathrm{C}_1-\mathrm{C}_3+\mathrm{C}_5-\mathrm{C}_7+\ldots \ldots=(\sqrt{2})^{\mathrm{n}} \sin \frac{\mathrm{n} \pi}{4}$
8. $\mathrm{C} _0+\mathrm{C} _4+\mathrm{C} _8+\mathrm{C} _{12}+\ldots \ldots=\frac{1}{2}\left(2^{\mathrm{n}-1}+(\sqrt{2})^{\mathrm{n}} \cos \frac{\mathrm{n} \pi}{4}\right) $
9. $\mathrm{C} _1+\mathrm{C} _5+\mathrm{C} _9+\mathrm{C} _{13}+\ldots \ldots=\frac{1}{2}\left(2^{\mathrm{n}-1}+(\sqrt{2})^{\mathrm{n}} \sin \frac{\mathrm{n} \pi}{4}\right) $
10. $\mathrm{C}_0+\mathrm{C} _3+\mathrm{C}_6+\mathrm{C} _9+\ldots \ldots=\frac{1}{3}\left(2^{\mathrm{n}}+2 \cos \frac{\mathrm{n} \pi}{3}\right) $
11. $\mathrm{C} _1+2 \mathrm{C}_2+\ldots \ldots \ldots \ldots .=\sum \mathrm{rC} _{\mathrm{r}}=\mathrm{n} \cdot 2^{\mathrm{n}-1}$
12. $\mathrm{C} _1-2 \mathrm{C} _2+3 \mathrm{C} _3 \ldots \ldots \ldots \ldots \ldots . \ldots=\quad \sum(-1)^{\mathrm{r}-1} \mathrm{rC} _{\mathrm{r}}=0 $
13. $1^2 \mathrm{C}_1+2^2 \mathrm{C}_2+\ldots \ldots \ldots \ldots \ldots \ldots=\mathrm{n}(\mathrm{n}+1) 2^{2 \mathrm{n}-2} $
14. $1^2 \mathrm{C}_1-2^2 \mathrm{C} _2 \ldots \ldots \ldots \ldots \ldots=0 $
15. $\mathrm{C} _0^2+\mathrm{C} _1^2+\mathrm{C} _2^2+\ldots \ldots \ldots \ldots . .={ }^{2 \mathrm{n}} \mathrm{C} _{\mathrm{n}}$
16. $C_ 0^2-C_ 1{}^2+C_2{}^2-C _3^2 \ldots \ldots \ldots \ldots \ldots=$
$\quad \left \{\begin{array}{l}0 \text{ if } n \text { is odd } \\ (-1)^{n/2}{}^n C_ {n/2} \text { if } n \text { is even }\end{array}\right.$
17. $\sum_{0 \leq i<j \leq n} C_i C_j=2^{2 n-1}-{ }^{2 n-1} C_n$
18. $\sum_{0 \leq i<j \leq n}\left(C_i-C_j\right)^2=(n+1){ }^{2 n} C_n-2^{2 n}$
Note : Consider the equation $\mathrm{x} _{1}+\mathrm{x} _{2}+\ldots \ldots \ldots . .+\mathrm{x} _{\mathrm{r}}=\mathrm{n}, . \mathrm{n} \in \mathrm{N}$.
Number of positive integral solutions $={ }^{n-1} C _{r-}$
Number of non negative integral solutions $={ }^{\mathrm{r}-1-1-1} \mathrm{C} _{\mathrm{r}-1}$
Solved examples
1. The value of $\sum _{\mathrm{r}=1}^{\mathrm{n}}{ }^{2 \mathrm{n}} \mathrm{C} _{\mathrm{r}} \mathrm{r}$ is
(a) $\mathrm{n} .2^{2 \mathrm{n}-1}$
(b) $2^{2 n-1}$
(c) $2^{\mathrm{n}-1}+1$
(d) None of these
Show Answer
Solution :
$ \begin{aligned} & \sum _{r=1}^{n} r \frac{2 n}{r}{ }^{2 n-1} C _{r-1} \\ = & 2 n \sum _{r=1}^{n}{ }^{2 n-1} C _{r-1} \\ = & 2 n\left({ }^{2 n-1} C _{0}+{ }^{2 n-1} C _{1}+{ }^{2 n-1} C _{2}+\ldots . .{ }^{2 n-1} C _{n-1}\right) \\ = & 2 n \cdot \frac{2^{2 n-1}}{2}=n \cdot 2^{2 n-1}\left[\because 2^{2 n-1}=C _{0}+C _{1}+\ldots \ldots . .+C _{2 n-1} \Rightarrow 2^{2 n-1}=2\left(C _{0}+C _{1}+C _{2}+\ldots \ldots+C _{n-1}\right)\right] \end{aligned} $
Answer: (a)
2. The coefficient of $x^{5}$ in the expansion of $(1+x)^{21}+(1+x)^{22}+\ldots \ldots \ldots .+(1+x)^{30}$ is
(a) ${ }^{31} \mathrm{C} _{5}{ }^{21} \mathrm{C} _{5}$
(b) ${ }^{31} \mathrm{C} _{6}{ }^{-21} \mathrm{C} _{6}$
(c) ${ }^{30} \mathrm{C} _{6}-{ }^{20} \mathrm{C} _{6}$
(d) None of these
Show Answer
Solution :
Co-efficient of $\mathrm{x}^{5}$ in $(1+\mathrm{x})^{21}+(1+\mathrm{x})^{22+}$ $+(1+x)^{30}$
$=$ Co-efficient of $x^{5}$ in $\frac{(1+x)^{21}\left\{(1+x)^{10}-1\right\}}{(1+x)-1}$
$\Rightarrow$ coefficient of $x^{6}$ in $(1+x)^{31}-(1+x)^{21}$ is ${ }^{31} \mathrm{C} _{6}-{ }^{21} \mathrm{C} _{6}$
Answer: (b)
3. The number of distinct terms in the expansion of $(x+y-z)^{16}$ is
(a) 136
(b) 153
(c) 16
(d) 17
Show Answer
Solution :
Apply ${ }^{n+r-1} \mathrm{C} _{\mathrm{r}-1}$ to get number of terms ${ }^{16+3-1} \mathrm{C} _{3-1}={ }^{18} \mathrm{C} _{2}=153$
Answer: (b)
4. If I is the integral part of $(2+\sqrt{3})^{\mathrm{n}}$ and $\mathrm{f}$ is the fractional part. Then $(\mathrm{I}+\mathrm{f})(1-\mathrm{f})$ is equal to
(a) $0$
(b) $1$
(c) $\mathrm{n}$
(d) None of these
Show Answer
Solution :
Let $(2+\sqrt{3})^{\mathrm{n}}=\mathrm{I}+\mathrm{f}…..(1)$
and $(2-\sqrt{3})^{\mathrm{n}}=\mathrm{F}………(2)$
(1) + (2) gives
$2\left({ }^{n} C _{0} \cdot 2^{\mathrm{n}+\mathrm{n}} \mathrm{C} _{2} \cdot 2^{\mathrm{n}-2}(\sqrt{3})^{2}+\ldots \ldots \ldots\right)=\mathrm{I}+\mathrm{f}+\mathrm{F}$
$\Rightarrow \mathrm{I}+\mathrm{f}+\mathrm{F}$ is an even integer
$\Rightarrow \mathrm{I}+\mathrm{F}$ is an integer
$ \begin{aligned} & \Rightarrow \mathrm{f}+\mathrm{F}=1 \\ & \mathrm{~F}=1-\mathrm{f} \\ & \therefore(\mathrm{I}+\mathrm{F})(1-\mathrm{f})=(2+\sqrt{3})^{\mathrm{n}}(2-\sqrt{3})^{\mathrm{n}}=(4-3)^{\mathrm{n}}=1 \end{aligned} $
Answer: (b)
5. If the middle term of $(1+x)^{2 n}(n \in N)$ is the greatest term of the expansion, then the interval in which $\mathrm{x}$ lies is
(a) $\left[\frac{\mathrm{n}+1}{\mathrm{n}}, \frac{\mathrm{n}+2}{\mathrm{n}}\right]$
(b) $\left[\frac{\mathrm{n}-1}{\mathrm{n}}, \frac{\mathrm{n}+1}{\mathrm{n}}\right]$
(c) $\left[\frac{\mathrm{n}}{\mathrm{n}+1}, \frac{\mathrm{n}+1}{\mathrm{n}}\right]$
(d) None of these
Show Answer
Solution :
| $\mathrm{T} _{n} \leq \mathrm{T} _{n}$ | $\&$ | $\mathrm{T} _{\mathrm{n}+1} \geq \mathrm{T} _{\mathrm{n}+2}$ |
| $\Rightarrow \quad{ }^{2 n} C_{n-1} x^{n-1} \leq{ }^{2 n} C_n \cdot x^n$ | $\&$ | ${ }^{2 n} C_n \cdot x^n \geq{ }^{2 n} C_{n+1} x^{n+1}$ |
| $\Rightarrow \quad \frac{{ }^{2 n} C_{n-1}}{{ }^{2 n} C_n} \leq x$ | $\&$ | $\frac{{ }^{2 n} C_n}{{ }^{2 n} C_{n+1}} \geq x$ |
| $x \geq \frac{n}{2 n-n+1}$ | $\&$ | $x \leq \frac{n+1}{2 n-n}$ |
$\Rightarrow \quad x \in\left[\frac{n}{n+1}, \frac{n+1}{n}\right]$
Answer: (c)
6. If $ C_{0}, \mathrm{C} _{1}, \mathrm{C} _{2},………C_n$ are the binomial coefficients in expansion of $(1+\mathrm{x})^{\mathrm{n}}, \mathrm{n}$ being even, then
$\mathrm{C} _0+\left(\mathrm{C} _0+\mathrm{C}_1\right)+\left(\mathrm{C} _0+\mathrm{C} _1+\mathrm{C} _2\right)+\ldots \ldots \ldots+\left(\mathrm{C}_0+\mathrm{C}_1+\ldots \ldots+\mathrm{C} _{\mathrm{n}-1}\right)$ is equal to
(a) $n \cdot 2^{\mathrm{n}}$
(b) $n \cdot 2^{n-1}$
(c) $\mathrm{n} \cdot 2^{\mathrm{n}-2}$
(d) $n \cdot 2^{n-3}$
Show Answer
Solution:
$\therefore \mathrm{C} _{0}+\left(\mathrm{C} _{0}+\mathrm{C} _{1}\right)+$ $.+\left(\mathrm{C} _{0}+\mathrm{C} _{1}+\ldots \ldots . \mathrm{C} _{\mathrm{n}-2}\right)+\left(\mathrm{C} _{0}+\mathrm{C} _{1}+\right.$ $\left.\ldots . \mathrm{C} _{n-1}\right)$
$=\left(C _{n}\right)+\left(C _{n}+C _{n-1}\right)+$ $+\left(\mathrm{C} _{0}+\mathrm{C} _{1}+\right.$ ..$\left.+\mathrm{C} _{\mathrm{n}-2}\right)+\left(\mathrm{C} _{0}+\mathrm{C} _{1}+\right.$. $+\mathrm{C} _{\mathrm{n}-1}$ )
$=2^{n}+2^{n}+2^{n}+\ldots \frac{n}{2}$ times (Adding the terms equidistant from the begining and the end)
$=\frac{\mathrm{n}}{2} \cdot 2^{\mathrm{n}}=\mathrm{n} \cdot 2^{\mathrm{n}-1}$
Answer: (b)
7. The number of terms in the expansion of $\left(\mathrm{x}^{3}+\frac{1}{\mathrm{x}^{3}}+1\right)^{100}$ is
(a) $201$
(b) $200$
(c) $300$
(d) $100 \mathrm{c} _{3}$
Show Answer
Solution :
$\left(\mathrm{x}^{3}+\frac{1}{\mathrm{x}^{3}}+1\right)^{100}=\mathrm{C} _{0}+\mathrm{C} _{1}\left(\mathrm{x}^{3}+\frac{1}{\mathrm{x}^{3}}\right)+\mathrm{C} _{2}\left(\mathrm{x}^{3}+\frac{1}{\mathrm{x}^{3}}\right)^{2}+\ldots \ldots \ldots \ldots+\mathrm{C} _{100}\left(\mathrm{x}^{3}+\frac{1}{\mathrm{x}^{3}}\right)^{100}$
gives terms of $\mathrm{x}^{3}, \mathrm{x}^{6}, \ldots \ldots \ldots \ldots \ldots \mathrm{x}^{300}, \frac{1}{\mathrm{x}^{3}}, \frac{1}{\mathrm{x}^{6}}, \ldots \ldots \ldots \ldots \frac{1}{\mathrm{x}^{300}}$ and a constant term
$\therefore 201$ terms
Answer: (a)
Exercises
1. If $\mathrm{C} _{\mathrm{r}}$ stands for ${ }^{n} \mathrm{C} _{\mathrm{r}}$, then the sum of the series
$2 \frac{\left(\frac{\mathrm{n}}{2}\right) !\left(\frac{\mathrm{n}}{2}\right) !}{\mathrm{n} !}\left(\mathrm{C} _{0}{ }^{2}-2 \mathrm{C} _{1}{ }^{2}+3 \mathrm{C} _{2}{ }^{2}-\ldots \ldots \ldots+(-1)^{\mathrm{n}}(\mathrm{n}+1) \mathrm{C} _{\mathrm{n}}{ }^{2}\right)$ when $\mathrm{n}$ is an even positive integer, is equal to
(a) $(-1)^{\mathrm{n} / 2}(\mathrm{n}+2)$
(b) $(-1)^{\mathrm{n}}(\mathrm{n}+1)$
(c) $(-1)^{\mathrm{n} / 2}(\mathrm{n}+1)$
(d) None of these
Show Answer
Answer: (a)2. $\left(\begin{array}{c}30 \\ 0\end{array}\right)\left(\begin{array}{c}30 \\ 10\end{array}\right)-\left(\begin{array}{c}30 \\ 1\end{array}\right)\left(\begin{array}{c}30 \\ 11\end{array}\right)+\ldots \ldots \ldots+\left(\begin{array}{c}30 \\ 20\end{array}\right)\left(\begin{array}{l}30 \\ 30\end{array}\right)$ is equal to
(a) ${ }^{30} \mathrm{C} _{11}$
(b) ${ }^{60} \mathrm{C} _{10}$
(c) ${ }^{30} \mathrm{C} _{10}$
(d) ${ }^{65} \mathrm{C} _{55}$
Show Answer
Answer: (c)3. If $\mathrm{r}=0,1,2, \ldots \ldots . .10$, let $\mathrm{A} _{\mathrm{r}}, \mathrm{B} _{\mathrm{r}}$ and $\mathrm{C} _{\mathrm{r}}$ denote, respectively, the coefficent of $\mathrm{x}^{\mathrm{r}}$ in the expansions of $(1+x)^{10},(1+x)^{20}$ and $(1+x)^{30}$. Then $\sum _{r=1}^{10} A _{r}\left(B _{10} B _{r}-C _{10} A _{r}\right)$ is equal to
(a) $\mathrm{B} _{10}-\mathrm{C} _{10}$
(b) $\mathrm{A} _{10}\left(\mathrm{~B} _{10}^{2}-\mathrm{C} _{10} \mathrm{~A} _{10}\right)$
(c) $0$
(d) $\mathrm{C} _{10}-\mathrm{B} _{10}$
Show Answer
Answer: (d)4. Value of $2^{k}\left(\begin{array}{l}n \\ 0\end{array}\right)\left(\begin{array}{l}n \\ k\end{array}\right)-2^{k-1}\left(\begin{array}{l}n \\ 1\end{array}\right)\left(\begin{array}{l}n-1 \\ k-1\end{array}\right)+2^{k-2}\left(\begin{array}{l}n \\ 2\end{array}\right)\left(\begin{array}{l}n-2 \\ k-2\end{array}\right)-\ldots \ldots .+(-1)^{k}\left(\begin{array}{l}n \\ k\end{array}\right)\left(\begin{array}{c}n-k \\ 0\end{array}\right)$ is
(a) $\left(\begin{array}{l}\mathrm{n} \\ \mathrm{k}\end{array}\right)$
(b) $\left(\begin{array}{l}\mathrm{n}-1 \\ \mathrm{k}-1\end{array}\right)$
(c) $1$
(d) None of these
Show Answer
Answer: (a)5. $\sum _{\mathrm{r}=0}^{\mathrm{n}}(-1)^{\mathrm{r}}{ }^{\mathrm{n}} \mathrm{C} _{\mathrm{r}}\left(\frac{1}{2^{\mathrm{r}}}+\frac{3^{\mathrm{r}}}{2^{2 \mathrm{r}}}+\frac{7^{\mathrm{r}}}{2^{\mathrm{r}}}+\frac{15^{\mathrm{r}}}{2^{4 \mathrm{r}}}+\ldots \ldots \ldots .\right.$. up to $\mathrm{m}$ tems $\left.)\right)=$
(a) $\frac{2^{\mathrm{mn}}-1}{2^{\mathrm{mn}}\left(2^{\mathrm{n}}-1\right)}$
(b) $\frac{2^{m}-2^{n}}{m-n}$
(c) $1$
(d) None of these
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Answer: (a)6. If $\sum _{\mathrm{r}=0}^{2 \mathrm{n}} \mathrm{a} _{\mathrm{r}}(\mathrm{x}-2)^{\mathrm{r}}=\sum _{\mathrm{r}=0}^{2 \mathrm{n}} \mathrm{b} _{\mathrm{r}}(\mathrm{x}-3)^{\mathrm{r}}$ and $\mathrm{a} _{\mathrm{k}}=1$ for all $\mathrm{k} \geq \mathrm{n}$, then $\mathrm{b} _{\mathrm{n}}=$
(a) ${ }^{n} C _{n}$
(b) ${ }^{2 n+1} C _{n+1}$
(c) ${ }^{2 n+1} \mathrm{C} _{n}$
(d) None of these
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Answer: (b)7. If $(1+x)^{n}=C _{0}+C _{1} x+C _{2} x^{2}+\ldots \ldots \ldots+C _{n} x^{n}$, then the sum of the products of the $\mathrm{C} _{\mathrm{i}}{ }^{\prime} \mathrm{s}$ taken two at a time represented by $\sum \sum \mathrm{C} _{\mathrm{i}} \mathrm{C} _{\mathrm{j}}(0 \leq \mathrm{i}<\mathrm{j} \leq \mathrm{n})$ is equal to
(a) $2^{2 \mathrm{n}-1}$
(b) $2^{\mathrm{n}}-\frac{(2 \mathrm{n}) !}{2(\mathrm{n} !)^{2}}$
(c) $2^{2 \mathrm{n}-1}-\frac{(2 \mathrm{n}) !}{2(\mathrm{n} !)^{2}}$
(d) None of these
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Answer: (c)8. Given $\mathrm{s} _{\mathrm{n}}=1+\mathrm{q}^{+} \mathrm{q}^{2}+\ldots \ldots \ldots .+\mathrm{q}^{\mathrm{n}}$ and $\mathrm{S} _{\mathrm{n}}=1+\frac{\mathrm{q}+1}{2}+\left(\frac{\mathrm{q}+1}{2}\right)^{2}+\ldots \ldots \ldots+\left(\frac{\mathrm{q}+1}{2}\right)^{\mathrm{n}}, \mathrm{q} \neq 1$ then ${ }^{n+1} \mathrm{C} _{1}+{ }^{n+1} \mathrm{C} _{2} \mathrm{~S} _{1}{ }^{n+1} \mathrm{C} _{3} \mathrm{~S} _{2}+\ldots \ldots . .+{ }^{n+1} \mathrm{C} _{\mathrm{n}+1} \mathrm{~S} _{\mathrm{n}}=$
(a) $2^{\mathrm{n}} \mathrm{S} _{\mathrm{n}}$
(b) $\mathrm{S} _{\mathrm{n}}$
(c) $\frac{S _{n}}{2^{n}}$
(d) None of these
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Answer: (a)9. $\lim _{n \rightarrow \infty} \sum _{r=0}^{n}\left(\begin{array}{l}n \ r\end{array}\right) \frac{1}{(r+3) n^{r}}=$
(a) $e$
(b) $\mathrm{e}-1$
(c) $\mathrm{e}+1$
(d) $e-2$
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Answer: (d)10. The coefficient of $x^{8}$ is the expansion of $\left(1+\frac{x^{2}}{2 !}+\frac{x^{4}}{4 !}+\frac{x^{6}}{6 !}+\frac{x^{8}}{8 !}\right)^{2}$ is
(a) $\frac{1}{315}$
(b) $\frac{2}{315}$
(c) $\frac{1}{105}$
(d) $\frac{1}{210}$
Show Answer
Answer: (a)11. Match the following:
| Column I | Column II |
|---|---|
| (a) The sum of binomial coefficients of terms containing power of $x$ more than $x^{20}$ in $(1+x)^{41}$ is divisibile by | (p) $2^{39}$ |
| (b) The sum of binomial coefficients of rational terms in the expansion of $(1+\sqrt{2})^{42}$ is divisible by | (q) $2^{40}$ |
| (c) If $\left(x+\frac{1}{x}+x^{2}+\frac{1}{x^{2}}\right)^{21}=\mathrm{a} _{0} x^{-42}+a _{1} x^{-41}+a _{2} x^{-40}+\ldots \ldots . .+a _{82} x^{40}$, then $\mathrm{a} _{0}+\mathrm{a} _{2}+\ldots . .+\mathrm{a} _{82}$ is divisible by | (r) $ 2^{41}$ |
| (d) The sum of binomial coefficients of positive real terms in the expansion of $(1+i x)^{42}(x>0)$ is divisible by | (s) $ 2^{38}$ |
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Answer: a $\rarr$ p, b $\rarr$ r, c $\rarr$ r, d $\rarr$ q12. Read the passage and answer the questions that follow:
An equation $\mathrm{a} _{0}+\mathrm{a} _{1} \mathrm{x}+\mathrm{a} _{2} \mathrm{x}^{2}+\ldots \ldots . .+\mathrm{a} _{99} \mathrm{x}^{99}+\mathrm{x}^{100}=0$ has roots ${ }^{99} \mathrm{C} _{0},{ }^{99} \mathrm{C} _{1},{ }^{99} \mathrm{C} _{2}, \ldots \ldots . .{ }^{99} \mathrm{C} _{99}$.
(i). The value of $\mathrm{a} _{99}$ is
(a) $2^{98}$
(b) $2^{99}$
(c) $-2^{99}$
(d) None of these
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Answer: (c)(ii). The value of $\mathrm{a} _{98}$ is
(a) $\frac{2^{198}-{ }^{198} \mathrm{C} _{99}}{2}$
(b) $\frac{2^{198}+{ }^{198} \mathrm{C} _{99}}{2}$
(c) $ 2{ }^{99}-{ }^{99} \mathrm{C} _{49}$
(d) None of these
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Answer: (a)(iii). The value of $\left({ }^{99} \mathrm{C} _{0}\right)^{2}+\left({ }^{99} \mathrm{C} _{1}\right)^{2}+$ $+\left({ }^{99} \mathrm{C} _{99}\right)^{2}$ is
(a) $2 \mathrm{a} _{98}-\mathrm{a} _{99}^{2}$
(b) $\mathrm{a} _{98}^{2}-\mathrm{a} _{99}^{2}$
(c) $\mathrm{a} _{99}^{2}-2 \mathrm{a} _{98}$
(d) None of these
