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

Question

The Number of ways of choosing $ 10 $ objects out of $ 31 $ objects of which $ 10 $ are identical and the remaining $ 21 $ are distinct, is:, $ 2^{20} $, $ 2^{21} $, $ 2^{20}+1 $, $ 2^{20}-1 $

MARKS App · Accepted Answer

There is only one way in which we can choose one, two or more objects from 10 identical objects. Number of ways of selecting 10 objects = 21 C 10 + 21 C 9 + 21 C 8 + 21 C 7 + 21 C 6 + 21 C 5 + 21 C 4 + 21 C 3 + 21 C 2 + 21 C 1 + 21 C 0 We know n C r = n C n - r and ⇒ 2 21 = 21 C 0 + 21 C 1 + 21 C 2 + 21 C 3 + 21 C 4 + . . . . . . . . . + 21 C 21 ∵ 21 C 21 = 21 C 0 21 C 20 = 21 C 1 21 C 19 = 21 C 2 = 2 21 C 0 + 21 C 1 + 21 C 2 + … … … … … + 21 C 10 ⇒ 21 C 0 + 21 C 1 + 21 C 2 + . . . . . . . . . + 21 C 10 = 2 20

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