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Choose whether the given statement is true or false.

n $C$ r + n $C$ r – 1 = n + 1 $C$ r

see full answer

21% of IItians & 23% of AIIMS delhi doctors are from Sri Chaitanya institute !!

An Intiative by Sri Chaitanya

True

False

answer is A.

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Detailed Solution

To check: n

C

r + n

C

r – 1 = n + 1

C

r
We know that,
n

C

= \frac{n!}{r! (n - r)!}

Now, LHS: n

C

r + n

C

r – 1

= \frac{n!}{r! (n - r)!}

\frac{n!}{(r - 1)! (n - r + 1)!}

= \frac{n!}{r . (r - 1)! (n - r)!}

\frac{n!}{(r - 1)! (n - r + 1) (n - r)!}

= \frac{n!}{(r - 1)! (n - r)!} (\frac{1}{r} + \frac{1}{n - r + 1})

= \frac{n!}{(r - 1)! (n - r)!} (\frac{n - r + 1 + r}{r (n - r + 1)})

= \frac{n!}{(r - 1)! (n - r)!} (\frac{n + 1}{r (n - r + 1)})

= \frac{(n + 1)!}{r! (n - r + 1)!}

=

n + 1

C

r
Thus, the given statement is true.

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