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

Aicharya · 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

\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}

The Number of ways of choosing 10 objects out of 31 objects of which 10 are identical and the remaining 21…