Multinomial Theorem
Multinomial Theorem
Let x1, x2, …….., xm be integers. Then number of solutions to the equation x1 + x2 +…….. + xm = n .....(i)
Subject to the condition
a1 ≤ x1 ≤ b1, a2 ≤ x2 ≤ b2, ……, am ≤ xm ≤ bm .....(ii)
is equal to the coefficient of xn in
This is because the number of ways, in which sum of m integers in (i) equals n, is the same as the number of times xn comes in (iii).
(1) Use of solution of linear equation and coefficient of a power in expansions to find the number of ways of distribution : (i) The number of integral solutions of x1 + x2 + x3 +…….. + xr = n where x1 ≥ 0, x2 ≥ 0, …….., xr ≥ 0 is the same as the number of ways to distribute n identical things among r persons.
This is also equal to the coefficient of xn in the expansion of (x0 + x1 + x2 + x3 + ……)r
