Question

Find n,if nC5=nC7 Answer it.....

Answer 1

GIVEN :–

$\\ \sf \implies {}^{n} c_{5} ={}^{n} c_{7}$

TO FIND :–

• Value of 'n' = ?

SOLUTION :–

$\\ \sf \implies {}^{n} c_{5} ={}^{n} c_{7}$

• We know that –

$\\ \large \longrightarrow \: { \boxed{ \sf {}^{n} c_{r} = \dfrac{n {\displaystyle !\,}}{(n - r){\displaystyle !\,}(r){\displaystyle !\,}} }}$

$\\ \sf \implies \dfrac{n {\displaystyle !\,}}{(n - 5){\displaystyle !\,}(5){\displaystyle !\,}} =\dfrac{n {\displaystyle !\,}}{(n - 7){\displaystyle !\,}(7){\displaystyle !\,}}$

$\\ \sf \implies \dfrac{1}{(n - 5){\displaystyle !\,}(5){\displaystyle !\,}} =\dfrac{1}{(n - 7){\displaystyle !\,}(7){\displaystyle !\,}}$

• We know that Value of Factorial –

$\\ \sf \longrightarrow \: n{\displaystyle !\,} = n.(n - 1).(n - 2) .(n - 3).........3.2.1$

• So , We can write this as –

$\\ \sf \implies \dfrac{1}{(n - 5)(n - 6)(n - 7){\displaystyle !\,}(5){\displaystyle !\,}} =\dfrac{1}{(n - 7){\displaystyle !\,}(7)(6)(5){\displaystyle !\,}}$

$\\ \sf \implies \dfrac{1}{(n - 5)(n - 6)} =\dfrac{1}{(7)(6)}$

$\\ \sf \implies (n - 5)(n - 6) =(7)(6)$

$\\ \sf \implies {n}^{2} - 6n - 5n + 30 =42$

$\\ \sf \implies {n}^{2} - 6n - 5n + 30 - 42 = 0$

$\\ \sf \implies {n}^{2} -11n - 12 = 0$

• Now Splitting Middle term –

$\\ \sf \implies {n}^{2} -12n + n - 12 = 0$

$\\ \sf \implies n(n - 12) + 1(n - 12) = 0$

$\\ \sf \implies (n + 1)(n - 12)= 0$

$\\ \sf \implies n = - 1 \: , \: 12$

$\\ \sf \: \: \because \: \: n = - 1 \: \: is \: \: not \: \: possible.$

• So that –

$\\ \: \: \longrightarrow \: \large{ \boxed{ \sf n = 12 }}$