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Multinomial Theorem

Note: Multinomial theorem is not included in JEE syllabus.

In the expansion, (a+x)n=............... there are a and x - two of them - hence the name binomial expansion. If we have (x1+x2+x3+........+xm)n expansion, then this expansion is called multinomial expansion.

\((x_1+x_2+x_3+........... +x_m )^n =\sum\limits_{n_1,n_2,n_3,......,n_m}{{{n!}\over {n_1!n_2!n_3!........n_m!}}x_1^{n_1} x_2^{n_2} x_3^{n_3} ........x_m^{n_m}\),

The summation is taken over all combinations of the indices n1 through nm such that n1 + n2 + n3 + ... + nm = n.

The numbers, \(\frac{n!}{n_1!n_2!n_3!........n_m!}\) are called multinomial coefficients.

Example, (a+b+c)2=a2+b2+c2+2ab+2bc+2ca.

\((a + b + c)^2 ={{2!}\over {2!}}a^2 +{{2!}\over {2!}}b^2 +{{2!}\over {2!}}c^2 + {{2!}\over {1!1!}}a^1 b^1 +{{2!}\over {1!1!}}b^1 c^1 +{{2!}\over {1!1!}}c^1 a^1\)

Application in combinations: The multinomial coefficients are the number of ways of depositing n distinguished objects in m baskets, with n1 in the first, and so on.

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