The number of ways in which $n$ different thing can be arranged into $r$ different groups if blank are allowed is

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Solution

$\begin{matrix} n d i f f e r e n t t h i n g s G_{1}, G_{2}, G_{3} . . . . . . . . . . . G_{r} T o t a l n o . o f w a y s =^{n + r - 1} C_{r - 1} \end{matrix}$

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The number of ways in which n different thing can be arranged into r different groups if blank are allowed is

ndifferentthingsG1,G2,G3...........GrTotalno.ofways=n+r−1Cr−1

The number of ways in which $n$ different thing can be arranged into $r$ different groups if blank are allowed is

$\begin{matrix} n d i f f e r e n t t h i n g s G_{1}, G_{2}, G_{3} . . . . . . . . . . . G_{r} T o t a l n o . o f w a y s =^{n + r - 1} C_{r - 1} \end{matrix}$