If for some m , n ; C m 6 + 2 ( C m + 1 6 ) + 6 C m + 2 > 8 C 3 and P 3 n - 1 : n P 4 = 1 : 8 , then P m + 1 n + n + 1 C m is equal to [2024]

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Solution

Q.
If for some $m, n;$ ${}^{6}{C_{m}} + 2 ({}^{6}{C_{m + 1}}) +^{6} C_{m + 2} >^{8} C_{3}$ and ${}^{n - 1}{P_{3}} :^{n} P_{4} = 1 : 8,$ then ${}^{n}{P_{m + 1}} +^{n + 1} C_{m}$ is equal to [2024]

1 372

2 384

3 380

4 376

Ans.
(1)

Given, ${}^{6}{C_{m}} + 2 ({}^{6}{C_{m + 1}}) +^{6} C_{m + 2} >^{8} C_{3}$

$\Rightarrow^{6} C_{m} +^{6} C_{m + 1} +^{6} C_{m + 1} +^{6} C_{m + 2} >^{8} C_{3}$

$\Rightarrow^{7} C_{m + 1} +^{7} C_{m + 2} >^{8} C_{3} \Rightarrow^{8} C_{m + 2} >^{8} C_{3} {or}^{8} C_{6 - m} >^{8} C_{3}$

$\Rightarrow m + 2 > 3$ i.e., $m > 1$

or $6 - m > 3$ i.e., $m < 3$ i.e., $1 < m < 3 \Rightarrow m = 2$

$∴$ ${}^{n}{P_{m + 1}} +^{n + 1} C_{m} =^{8} P_{3} +^{9} C_{2} = 336 + 36 = 372$

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