JEE Main & Advanced Mathematics Permutations and Combinations Multinomial Theorem

Let \[{{x}_{1}},\,{{x}_{2}},\,.......,{{x}_{m}}\] be integers. Then number of solutions to the equation \[{{x}_{1}}+{{x}_{2}}+......+{{x}_{m}}=n\]                                                                                                         .....(i)

Subject to the condition

\[{{a}_{1}}\le {{x}_{1}}\le {{b}_{1}},\,{{a}_{2}}\le {{x}_{2}}\le {{b}_{2}},.......,{{a}_{m}}\le {{x}_{m}}\le {{b}_{m}}\]          .....(ii)

is equal to the coefficient of \[{{x}^{n}}\]in

\[({{x}^{{{a}_{1}}}}+{{x}^{{{a}_{1}}+1}}+......+{{x}^{{{b}_{1}}}})\,({{x}^{{{a}_{2}}}}+{{x}^{{{a}_{2}}+1}}+.....+{{x}^{{{b}_{2}}}})......({{x}^{{{a}_{m}}}}+{{x}^{{{a}_{m+1}}}}+.....+{{x}^{{{b}_{m}}}})\].....(iii)

This is because the number of ways, in which sum of m integers in (i) equals n, is the same as the number of times \[{{x}^{n}}\] comes in (iii).

(1) Use of solution of linear equation and coefficient of a power in expansions to find the number of ways of distribution : (i) The number of integral solutions of \[{{x}_{1}}+{{x}_{2}}+{{x}_{3}}+......+{{x}_{r}}=n\] where \[{{x}_{1}}\ge 0,\,{{x}_{2}}\ge 0,\,......{{x}_{r}}\,\ge 0\] is the same as the number of ways to distribute n identical things among r persons.

This is also equal to the coefficient of \[{{x}^{n}}\] in the expansion of \[{{({{x}^{0}}+{{x}^{1}}+{{x}^{2}}+{{x}^{3}}+......)}^{r}}\]

= coefficient of \[{{x}^{n}}\] in \[{{\left( \frac{1}{1-x} \right)}^{r}}\]

= coefficient of \[{{x}^{n}}\] in \[{{(1-x)}^{-r}}\]

= coefficient of \[{{x}^{n}}\] in

\[\left\{ 1+rx+\frac{r(r+1)}{2!}{{x}^{2}}+....+\frac{r(r+1)\,(r+2)....(r+n-1)}{n\,!} \right.{{x}^{n}}+......\]

\[=\frac{r(r+1)\,(r+2)....(r+n-1)}{n\,!}=\frac{(r+n-1)!}{n!(r-1)!}{{=}^{n+r-1}}{{C}_{r-1}}\].

(2) The number of integral solutions of \[{{x}_{1}}+{{x}_{2}}+{{x}_{3}}+\] \[.....+{{x}_{r}}=n\] where \[{{x}_{1}}\ge 1,\,{{x}_{2}}\ge 1,.......{{x}_{r}}\ge 1\] is same as the number of ways to distribute n identical things among r persons each getting at least 1. This also equal to the coefficient of \[{{x}^{n}}\] in the expansion of \[{{({{x}^{1}}+{{x}^{2}}+{{x}^{3}}+......)}^{r}}\].

= coefficient of \[{{x}^{n}}\] in \[{{\left( \frac{x}{1-x} \right)}^{r}}\]

= coefficient of \[{{x}^{n}}\] in \[{{x}^{r}}{{(1-x)}^{-r}}\]

= coefficient of \[{{x}^{n}}\] in

    \[{{x}^{r}}\left\{ 1+rx+\frac{r\,(r+1)}{2\,!}{{x}^{2}}+...+\frac{r\,(r+1)\,(r+2)...(r+n-1)}{n\,!}{{x}^{n}}+.. \right\}\]

= coefficient of \[{{x}^{n-r}}\] in

\[\left\{ 1+rx+\frac{r(r+1)}{2!}{{x}^{2}}+...+\frac{r(r+1)(r+2)....(r+n-1)}{n\,!}{{x}^{n}}+..... \right\}\]

= \[\frac{r(r+1)\,(r+2)......(r+n-r-1)}{(n-r)\,!}\] = \[\frac{r(r+1)\,(r+2).....(n-1)}{(n-r)!}\] = \[\frac{(n-1)\,!}{(n-r)!(r-1)!}{{=}^{n-1}}{{C}_{r-1}}\].