# Permutation and Combination problem : In how many ways can Rs. 16 be divided into 4 person when none of them get ...

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## Solution 1

The easiest way to solve it is to imagine giving each person Rs. $3$ right away and then distributing the remaining Rs. $4$ arbitrarily. In other words, reduce the problem to distributing Rs. $4$ amongst $4$ people with no restrictions: you have the formula for that, with $n=r=4$:

$$\binom{4+4-1}{4-1}=\binom73=35\;.$$

## Solution 2

What you want is, how many ways are there to write $$\{1, \ldots, 16\} = A_1 \dot{\cup} A_2 \dot\cup A_3 \dot\cup A_4$$ Under the constraint, that $|A_i| \geq 3 \quad \forall\ i$.
Did I understand that correctly?

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### Sachin

Updated on August 01, 2022

I have this : The number of ways of distributing $n$ things all alike into $r$ different groups is $^{n+r-1}C_{r-1}$ lots , no lots being blank:
Coefficient of $x^n$ in $(1+x+x^2+.....)^r$