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

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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
$\Rightarrow 2^{21} =^{21} C_{0} +^{21} C_{1} +^{21} C_{2} +^{21} C_{3} +^{21} C_{4} + . . . . . . . . . +^{21} C_{21} \{\begin{matrix} ∵^{21} C_{21} =^{21} C_{0} \\ ^{21} C_{20} =^{21} C_{1} \\ ^{21} C_{19} =^{21} C_{2} \end{matrix}$
$= 2 (^{21} C_{0} +^{21} C_{1} +^{21} C_{2} + \dots \dots \dots \dots \dots +^{21} C_{10})$
$\Rightarrow^{21} C_{0} +^{21} C_{1} +^{21} C_{2} + . . . . . . . . . +^{21} C_{10} = 2^{20}$

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