Question 11 and 12, Exercise 6.3

Solutions of Question 11 and 12 of Exercise 6.3 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan.

Number of diagonals in $n$-sided polygon is $35$. Find the number $n$?

Solution.

So given that \begin{align*}{ }^{n} C_{2}-n&=35\\ \text{or} \quad \dfrac{n!}{2!(n-2)!}-n&=35\\ \dfrac{n(n-1)(n-2)!}{2(n-2)!}-n&=35\\ \dfrac{n(n-1)-2 n}{2}&=35\\ n^{2}-n-2 n&=70\\ n^{2}-3 n-70&=0\\ n^{2}+7 n-10 n-70 & =0 \\ n(n+7)-10(n+7) & =0 \\ (n+7)(n-10) & =0 \\ n+7 & =0 \\ \text { or } \quad n-10 & =0 \\ n & =-7 \\ n & =10 \end{align*} $n $ can not be negative, so $n =10$

For the post of $6$ officers, there are $100$ appliciants,
$2$ posts are reserved for serving candidiates and remaining for others.
There are $20$ serving candidates among the appliciants. In how many ways this selection can be made?

Solution.

$2$ candidate would be chosen out of $20$ applications of serving candidates
and $4$ candidates would be chosen out of remaining $80$ applicants.
Possible ways of selection
$={ }^{20} C_{2} \times{ }^{80} C_{4}=190 \times 1581580=300500200$

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