Question

In the expansion of ${\left(a+b\right)}^{\mathbf{n}}$(see Example 2), let $\mathit{a}\mathbf{=}\mathit{b}\mathbf{=}\mathbf{1}$, and interpret the terms of the expansion to show that the total number of combinations of n things taken1, 2, 3, · · · , n at a time, is ${\mathbf{2}}^{\mathbf{n}}\mathbf{-}\mathbf{1}$
.

Short answer

Answer

The total number of combinations of n things taken 1, 2, 3, · · · , n at a time, is $2^n-1$.

Step-by-step solution

Given Information

$a=b=1$in the expansion of${\left(a+b\right)}^n$.

Definition of Independent Event

When the order of arrangement is definite, the permutation is applied and when the order is not definite,combination is applied.

Find the total number of combinations of n things taken 1, 2, 3, · · ·, n at a time,

The expansion of ${\left(a+b\right)}^n$is ${\left(a+b\right)}^n=\sum _{r=0}^{n}C\left(n,r\right)a^{n-r}{b}^r$.

Substitute 1 for a and 1 for b into the expansion and simplify.

$$\begin{gathered} {\left(1+1\right)}^n=C\left(n,0\right)1^{n-0}{1}^0+C\left(n,2\right)1^{n-2}{1}^2+8+C\left(n,n\right)1^{n-n}{1}^n \\ {2}^n=C\left(n,0\right)+C\left(n,1\right)+C\left(n,2\right)+\otimes +C\left(n,n\right) \\ C\left(n,1\right)+C\left(n,2\right)+\otimes +C\left(n,n\right)=2^n-C\left(n,0\right) \\ C\left(n,1\right)+C\left(n,2\right)+\otimes +C\left(n,n\right)=2^n-1 \end{gathered}$$

From the obtained simplification, it can be observed that the total number of combinations of nthings taken 1, 2, 3, · · · , nat a time, is$2^n-1$.