SATHEE: Binomial Theorem 2 Question 4

Binomial Theorem 2 Question 4

4. For $r = 0, 1, \dots, 10$ , if $A_{r}, B_{r}$ and $C_{r}$ denote respectively the coefficient of $x^{r}$ in the expansions of $(1 + x)^{10}, (1 + x)^{20}$ and $(1 + x)^{30}$ . Then, $\sum_{r = 1}^{10} A_{r} (B_{10} B_{r} - C_{10} A_{r})$ is equal to

(a) $B_{10} - C_{10}$

(b) $A_{10} (B_{10}^{2} - C_{10} A_{10})$

(d) $C_{10} - B_{10}$

(2010)

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Answer:

Correct Answer: 4. (d)

Solution:

$A_{r} =$ Coefficient of $x^{r}$ in $(1 + x)^{10} =^{10} C_{r}$

$B_{r} =$ Coefficient of $x^{r}$ in $(1 + x)^{20} =^{20} C_{r}$

$C_{r} =$ Coefficient of $x^{r}$ in $(1 + x)^{30} =^{30} C_{r}$

$∴ \sum_{r = 1}^{10} A_{r} (B_{10} B_{r} - C_{10} A_{r}) = \sum_{r = 1}^{10} A_{r} B_{10} B_{r} - \sum_{r = 1}^{10} A_{r} C_{10} A_{r}$

$= \sum_{r = 1}^{10}^{10} C_{r}^{20} C_{10}^{20} C_{r} - \sum_{r = 1}^{10}^{10} C_{r}^{30} C_{10}^{10} C_{r}$

$= \sum_{r = 1}^{10}^{10} C_{10 - r}^{20} C_{10}^{20} C_{r} - \sum_{r = 1}^{10}^{10} C_{10 - r}^{30} C_{10}^{10} C_{r}$

$=^{20} C_{10} \sum_{r = 1}^{10}^{10} C_{10 - r} \cdot^{20} C_{r} -^{30} C_{10} \sum_{r = 1}^{10}^{10} C_{10 - r}^{10} C_{r}$

$=^{20} C_{10} (^{30} C_{10} - 1) -^{30} C_{10} (^{20} C_{10} - 1)$

$=^{30} C_{10} -^{20} C_{10} = C_{10} - B_{10}$

$∴ A =^{30} C_{0}^{30} C_{10} -^{30} C_{1}^{30} C_{11} +^{30} C_{2}^{30} C_{12}$

$=$ Coefficient of $x^{20}$ in $(1 + x)^{30} (1 - x)^{30}$

$- \dots +^{30} C_{20}^{30} C_{30}$

$=$ Coefficient of $x^{20}$ in ${(1 - x^{2})}^{30}$

$=$ Coefficient of $x^{20}$ in $\sum_{r = 0}^{30} (- 1)^{r^{30}} C_{r} {(x^{2})}^{r}$

$= (- 1)^{10}^{30} C_{10}$ [for coefficient of $x^{20}$ , put $r = 10$ ] $=^{30} C_{10}$

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Binomial Theorem 2 Question 4

4. For $r = 0, 1, \dots, 10$ , if $A_{r}, B_{r}$ and $C_{r}$ denote respectively the coefficient of $x^{r}$ in the expansions of $(1 + x)^{10}, (1 + x)^{20}$ and $(1 + x)^{30}$ . Then, $\sum_{r = 1}^{10} A_{r} (B_{10} B_{r} - C_{10} A_{r})$ is equal to

Answer:

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Binomial Theorem 2 Question 4

4. For r=0,1,…,10, if Ar,Br and Cr denote respectively the coefficient of xr in the expansions of (1+x)10,(1+x)20 and (1+x)30. Then, ∑r=110Ar(B10Br−C10Ar) is equal to

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चिकित्सा तैयारी

एनसीईआरटी पुस्तकें और समाधान

महत्वपूर्ण संसाधन

अन्य संसाधन

पाठ्यक्रम

परीक्षा मंच

नमूना मॉक टेस्ट

सदस्यता

एनसीईआरटी व्यायाम वीडियो समाधान

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4. For $r = 0, 1, \dots, 10$ , if $A_{r}, B_{r}$ and $C_{r}$ denote respectively the coefficient of $x^{r}$ in the expansions of $(1 + x)^{10}, (1 + x)^{20}$ and $(1 + x)^{30}$ . Then, $\sum_{r = 1}^{10} A_{r} (B_{10} B_{r} - C_{10} A_{r})$ is equal to