Question

A man has 10 friends. In how many ways he can invite one or more of them to a party?

• A. $10!$
• B. $2^{10}$
• C. $10!-1$
• D. $2^{10}-1$

Answer 1

The correct option is D

$2^{10}-1$

Explanation for the correct option:

Here, the number of friends $=10$

Number of ways of inviting one friend out of ten ${=}^{10}{\mathrm{C}}_1$

Hence, the number of ways of inviting one or more friends ${=}^{10}{\mathrm{C}}_1+{}^{10}{\mathrm{C}}_2+{}^{10}{\mathrm{C}}_3....+{}^{10}{\mathrm{C}}_{10}$ …(1)

We know, ${}^{\mathbf{n}}{\mathbf{C}}_{\mathbf{0}}\mathbf{}\mathbf{+}{\mathbf{}}^{\mathbf{n}}{\mathbf{C}}_{\mathbf{1}}\mathbf{}\mathbf{+}{\mathbf{}}^{\mathbf{n}}{\mathbf{C}}_{\mathbf{2}}\mathbf{}\mathbf{+}{\mathbf{}}^{\mathbf{n}}{\mathbf{C}}_{\mathbf{3}}\mathbf{}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}{\mathbf{+}}^{\mathbf{n}}{\mathbf{C}}_{\mathbf{n}}\mathbf{=}{\mathbf{2}}^{\mathbf{n}}$

$\mathbf{\Rightarrow }{\mathbf{}}^{\mathbf{n}}{\mathbf{C}}_{\mathbf{1}}\mathbf{}\mathbf{+}{\mathbf{}}^{\mathbf{n}}{\mathbf{C}}_{\mathbf{2}}\mathbf{}\mathbf{+}{\mathbf{}}^{\mathbf{n}}{\mathbf{C}}_{\mathbf{3}}\mathbf{}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{}{\mathbf{+}}^{\mathbf{n}}{\mathbf{C}}_{\mathbf{n}}\mathbf{=}\mathbf{}{\mathbf{2}}^{\mathbf{n}}\mathbf{}\mathbf{-}\mathbf{}\mathbf{1}$ (2)

Now comparing the equations (1) and (2) we get, $n=10$

$\Rightarrow {}^{10}{\mathrm{C}}_1+{}^{10}{\mathrm{C}}_2+{}^{10}{\mathrm{C}}_3....+{}^{10}{\mathrm{C}}_{10}=2^{10}-1$

Therefore, the man has $2^{10}-1$ ways of inviting his friends.

Hence, Option (D) is correct.