there are 3 friends a,b,c they all have some 1rupee coins. a has 6 and b and c have 7 and 8 cons respectively. they want to donate rs 10 to a trust . in how many ways they can do it

pls explain using multinomial theorem and pls tell what is multinomial theorem.

So we are given

x+y+z=10

x can range from 0-6 since a can choose 0 to 6 coins

similarly

y can range from 0-7

z can range from 0-8

The number of ways in which they can donate money

$=({x}^{0}+{x}^{1}+{x}^{2}+{x}^{3}+{x}^{4}+{x}^{5}+{x}^{6})({x}^{0}+{x}^{1}+{x}^{2}+{x}^{3}+{x}^{4}+{x}^{5}+{x}^{6}+{x}^{7})({x}^{0}+{x}^{1}+{x}^{2}+{x}^{3}+{x}^{4}+{x}^{5}+{x}^{6}+{x}^{7}+{x}^{8})\phantom{\rule{0ex}{0ex}}=\frac{({x}^{7}-1)}{(x-1)}\frac{({x}^{8}-1)}{(x-1)}\frac{({x}^{9}-1)}{(x-1)}\phantom{\rule{0ex}{0ex}}=(x7-1)(x8-1)(x9-1)(x-1)-3=(1-x7)(1-x8)(1-x9)(1-x)-3=(1-x7)(1-x8)(1-x9)(1+3$

_{c1}

_{c2}

^{2}

_{c10}

^{10}

^{10}

^{10}

_{c10}

_{c3}

_{c2}

_{c1}

**
**