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.
x+y+z = 10
x,y,z are the number of coins deonated by a,b,c respectively.
x can be any value from 0 to 6 , because 'a' can donate 0 coins to 6 coins .
Similarly, y and z are from '0 to 7' and '0 to 8' respectively.
So, the coefficient of x10 in the expression (given below) are the no. of ways in which they can donate 10 coins.
(x0+x1+x2+ x3+x4+x5+x6)(x0+x1+ x2+x3+x4+ x5+x6+x7)(x0+x1+x2+ x3+x4+x5+ x6+x7+x8)
=[1(x7-1)/(x-1)] .[1(x8-1)/(x-1)] .[1(x9-)/(x-1)]
=(1-x7)(1-x8)(1-x9) (1+ 3C1x+ 4C2x2+ 5C3x3+ 6C4x4+.......+ 12C10x10+...........) [because, (1-x)-n=1+ nC1x + n+1C2x2+ n+2C3x+ n+3C4x+ n+4C5+........]
= (12C10 -5C3 -4C2 -3C1)x10+...... other terms.
= 66 -10 -6 -3
= 47 ways
I hope, it helps you.