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Prove that in the expansion of (1+x)^n(1...

Prove that in the expansion of `(1+x)^n(1+y)^n(1+z)^n` , the sum of the coefficients of the terms of degree `ri s^(3n)C_r` .

Text Solution

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The given expansion can be written as
`underset("n factors")({(1+x)(1+x)(1+x)"......"(1+x)})underset("n factors")({(1+y)(1+y)(1+y)"......"(1+y)})underset("n factors")({(1+z)(1+z)(1+z)"......"(1+z)})`
There are 3n factors in this product. To get a term of degree r, we choose brackets out of these 3n brackest and then multiply second term in each bracket. There are `.^(3n)C_(r )` such terms each having the coefficient 1. Hence the sum of the coefficient is `.^(3n)C_(r )`.
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