Suppose and are positive integers such that the two consecutive binomial coefficients…

MATHEMATICS · Multi-part · CBSE · Mathematics · Binomial Theorem · Binomial Theorem

Problem

Suppose

n

and

r

are positive integers such that the two consecutive binomial coefficients

{}^{n}C_{r}

and

{}^{n}C_{r-1}

are equal. Answer the following questions. (i) Show that if

{}^{n}C_{r} = {}^{n}C_{r-1}

, then

2r = n+1

. (ii) Using the relation obtained in part (i), explain why no solution in positive integers

r

exists when

n

is even. (iii) Give one concrete numerical example of

(n,r)

that satisfies

{}^{n}C_{r} = {}^{n}C_{r-1}

and verify the equality by direct calculation.