Question

Find the number of ways in which $16$ identical things can be distributed among $4$ persons, it each person gets at least $3$ things.

Answer 1

Let a,b,c,d be the numbers of things distributed

to each person such that

a+b+c+d=16.

we distribute 3 things to each person ,then

$3\times 4=12$ things are distributed . This only 4 things is

left.

we know that numbers of ways of distributing

n identical objects among r groups ${}^{n+r-1}{C}_{r-1}$

Hence numbers of ways of distributing 4 identical things

among 4 person ${=}^{4+4-1}{C}_{4-1}{=}^7{C}_3=\frac{7\times 6\times 5}{1\times 2\times 3}$

=35

Hence numbers of ways of distributing 16 dental things

among 4 person such that each person gets atleast

3 Things =35.