Least value of natural number n satisfying C(n,,5) + C(n,,6),, > C(n + 1,,5) is

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6.Permutation and Combination

easy

The least value of natural number $n$ satisfying $C(n,\,5) + C(n,\,6)\,\, > C(n + 1,\,5)$ is

A

$11$

B

$10$

C

$12$

D

$13$

Solution

(a) $^n{C_5}\, + {\,^n}{C_6}\,\, > \,{\,^{n + 1}}{C_5}$ ==> $^{n + 1}{C_6}\,\, > \,{\,^{n + 1}}{C_5}$

==> $\frac{{(n + 1)!}}{{6!\,.\,(n – 5)!}}\,.\,\frac{{5!\,.\,(n – 4)!}}{{(n + 1)!}}\,\, > \,\,1$

==> $\frac{{(n – 4)}}{6}\,\, > \,\,1$

==> $n – 4\,\, > \,\,6\,\,\,\, \Rightarrow \,\,\,n\,\,\, > \,\,10$

Hence according to options n = 11.

Standard 11

Mathematics

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