Question

please solve this question from permutation and combination.​

Answer 1

Given Question :- What is the value of

$\rm \: ^{16}C_{1} \: + \: ^{15}C_{1} \: + \: ^{14}C_{1} \: + \: ^{14}C_{2}$

$\: \: \: \rm \: (a) \: ^{15}C_{2}$

$\: \: \: \rm \: (b) \: ^{16}C_{2}$

$\: \: \: \rm \: (c) \: ^{17}C_{3}$

$\: \: \: \rm \: (d) \: ^{17}C_{15}$

$\large\underline{\sf{Solution-}}$

Given expression is

$\rm \: ^{16}C_{1} \: + \: ^{15}C_{1} \: + \: ^{14}C_{1} \: + \: ^{14}C_{2}$

can be regrouped as

$\rm \: = \: ^{16}C_{1} \: + \: ^{15}C_{1} \: + (\: ^{14}C_{1} \: + \: ^{14}C_{2} \: )$

We know,

$\boxed{ \rm{ \:\bf\: ^{n}C_{r} \: + \: ^{n}C_{r - 1} \: = \: ^{n + 1}C_{r} \:}}$

So, using this result, we get

$\rm \: = \: ^{16}C_{1} \: + \: ^{15}C_{1} \: + (\: ^{15}C_{2} \: )$

$\rm \: = \: ^{16}C_{1} \: + \: ^{15}C_{1} \: + \: ^{15}C_{2} \:$

can be regrouped as

$\rm \: = \: ^{16}C_{1} \: + ( \: ^{15}C_{1} \: + \: ^{15}C_{2} \: )$

$\rm \: = \: ^{16}C_{1} \: + ( \: ^{16}C_{2} \: )$

$\rm \: = \: ^{16}C_{1} \: + \: ^{16}C_{2} \:$

$\rm \: = \: ^{16 + 1}C_{2} \:$

$\rm \: = \: ^{17}C_{2} \:$

We know,

$\boxed{ \rm{ \:^{n}C_{r} \: = \: ^{n}C_{n - r} \: }}$

So, using this result, we get

$\rm \: = \: ^{17}C_{17 - 2}$

$\rm \: = \: ^{17}C_{15}$

Hence,

$\rm\implies \bf\: ^{16}C_{1} \: + \: ^{15}C_{1} \: + \: ^{14}C_{1} \: + \: ^{14}C_{2} = \: ^{17}C_{15}$

So, Option (d) is correct