JEE Main · 2019 · 12 Apr 2019 · Shift 1 · Permutations and Combinations

The number of ways of choosing 10 objects out of 31 objects of which 10 are identical and the remaining 21 are distinct, is:

Answer: (A)

Solution

To pick

10

objects from

10

identical ones and

21

distinct ones, choose

k

of the identical objects (which can be done in just one way for each

k

) and the remaining

10 - k

from the distinct ones, so the count is

\sum_{k = 0}^{10} (10 - k 21) = \sum_{j = 0}^{10} (j 21)

. Since

21

is odd, the

22

binomial coefficients

(0 21), \dots, (21 21)

split into two equal halves, so

\sum_{j = 0}^{10} (j 21)

is exactly half of

2^{21}

, namely

2^{20}

JEE Main · 2019 · 12 Apr 2019 · Shift 1 · Single correct option