Page 187 - DMTH202_BASIC_MATHEMATICS_II

Basic Mathematics-II

Notes In analogy with ‘binomial’, which is a sum of two symbols, we have ‘multinomial’ which is a
sum of two or more (though finite) distinct symbols. Specifically we will consider the expansion
n
of (a +a + …+ a ) .
1 2 m

Notes For the expansion we can use the same technique as we use for the binomial
expansion. We consider the nth power of the multinomial as the product of n factors, each
of which is the same multinomial.
Every term in the expansion can be obtained by picking one symbol from each factor and
multiplying them. Clearly, any term will be of the form where r , r ,…, are non-negative
1 2
integers such that r + r +…+ r = n. Such a term is obtained by selecting a from r factors, a
1 2 m 1 1 2
from r factors from among the remaining (n – r ) factors, and so on. This can be done in:
2 1
C(n, r )  C(n – r , r )  C(n – r – r , r ) …C(n – r – r …………… – r , r ) ways.
1 1 2 1 2 3 1 2 m – 1 m
! n
If you simplify this expression, it will reduce to .
r
r 1 ! !... !
r
2
m
So, we see that the multinomial expansion is
! n
n
a
( + + ... + a ) =  a 1 r ,a 2 r ...a m r
a
1 2 m 1 2 m
r ! !... !
r
r
1 2 m
where the summation is over all non-negative integers r , r ,…, r adding to n.
1 2 m
! n
m r
1 r
,
The coefficient of a a 2 2 r ...a in the expansion (a + a +… + a ) is r 1 ! !... ! , and is called a
n
r
r
m
1
2
m
1
m
2
multinomial coefficient, in analogy with the binomial coefficient. We represent this by C(n, r ,
1
 n 
r …, r ). This is also represented by many authors as   .
r
2 m r , ,...r
 1 2 m 
10
2 2 2 2 2
For instance, the coefficient of x y z t u in the expansion of (x + y + z + u) is C(10; 2, 2, 2, 2, 2) =
5
10!/(2!) .
Did u know? Multinomial expansion refers to the expansion of a positive integral power of
a multinomial.
Self Assessment
Fill in the blanks:
8. A .............................. refers to the expansion of a positive integral power of such a binomial.
9. In analogy with ‘binomial’, which is a sum of two symbols, we have ‘multinomial’ which
is a sum of two or more (though finite) .............................. symbols.
n
10. The coefficient of a b n “ r in the expansion of (a + b) is .............................. .
r
11. .............................. expansion refers to the expansion of a positive integral power of a
multinomial.
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