Question

(20 ) If 77Cr is maximum then r = ………….

(A) 38 (B) 38.5 (C) 39 (D) 4

Answer 1

SOLUTION

TO CHOOSE THE CORRECT OPTION

$\sf{ {}^{77}C_r \: \: is \: maximum \: when \: \: r = }$

(A) 38

(B) 38.5

(C) 39

(D) 4

CONCEPT TO BE IMPLEMENTED

We know that

When n is even

$\displaystyle \sf{ {}^{n}C_r \: \: is \: maximum \: when \: \: r = \frac{n}{2} }$

When n is odd

$\displaystyle \sf{ {}^{n}C_r \: \: is \: maximum \: when \: \: r = \frac{n - 1}{2} \: \: and \: \: r = \frac{n + 1}{2} }$

EVALUATION

Here the given expression is

$\sf{ {}^{77}C_r \: }$

So n = 77 which is odd

Hence

$\sf{ {}^{77}C_r \: \: is \: maximum \: when \: \: }$

$\displaystyle \sf{r = \frac{77 - 1}{2} \: \: and \: \: \frac{77 + 1}{2} }$

$\displaystyle \sf{ \implies \: r = \frac{76}{2} \: \: and \: \: \frac{78}{2} }$

$\displaystyle \sf{ \implies \: r = 38 \: \: and \: \: 39}$

Since

$\displaystyle \sf{ {}^{n}C_r = {}^{n}C_{n - r}}$

$\implies \displaystyle \sf{ {}^{77}C_{38} = {}^{77}C_{77 - 38}}$

$\implies \displaystyle \sf{ {}^{77}C_{38} = {}^{77}C_{39}}$

Hence the correct options are

(A) 38 and (C) 39

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