SATHEE: Permutations and Combinations 2 Question 25

Permutations and Combinations 2 Question 25

25. If $^{n} C_{r - 1} = 36,^{n} C_{r} = 84$ and $^{n} C_{r + 1} = 126$ , then find the values of $n$ and $r$ .

$(1979, 3$ M)

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Answer:

Correct Answer: 25. ( $n = 9 a n d r = 3$ )

Solution:

We know that, $\frac{^{n} C_{r}}{^{n} C_{r - 1}} = \frac{n - r + 1}{r}$

$\begin{array}{ll} \Rightarrow & \frac{84}{36} = \frac{7}{3} = \frac{n - r + 1}{r} \\ \Rightarrow & 3 n - 10 r + 3 = 0 \end{array}$

Also given, $\frac{^{n} C_{r}}{^{n} C_{r + 1}} = \frac{84}{126}$

$\begin{aligned} \Rightarrow & \frac{r + 1}{n - r} & = \frac{2}{3} \\ \Rightarrow & 2 n - 5 r - 3 & = 0 \end{aligned}$

On solving Eqs. (i) and (ii), we get

$r = 3 and n = 9$

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