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Multinomial Expansions

(x1 + x2 + ... + xr)n
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where P1 + P2 + P3 + ... + Pr = n and 0 ≤ P1, P2, ..., Prn.
 
Number of terms in the expansion in (x1 + x2 + ... + xr)n is equal to number non-negative integral solution of equation x1 + x2 + ... + xr = n.

Sum of coefficient in the expansion

For (x + y)n = nC0xn + nC1xn–1y + nC2 xn–2 y2 + ... + nCnyn we get the sum of coefficient by putting x = y = 1, which is 2n.
 
Similarly in the expansion of (x + y + z)n we get the sum of coefficient by putting x = y = z = 1.
For expansion of the type (x2 + x + 1)n = a0 + a1x + a2x2 + ... + a2nx2n we get the sum of coefficient by putting x = 1 or 3n = a0 + a1 + a2 + ... + a2n which is required sum of coefficient.
 
In the above expansion to get the sum of coefficient of even powers of x and odd powers of x, put x = 1 and x = –1 alternatively and then add or subtract the two results.




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