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Question
In the expansion of (1+x)n(1+y)n(1+z)n, the sum of the coefficients of the terms of degree r is
In the expansion of (1+x)n(1+y)n(1+z)n, the sum of the coefficients of the terms of degree r is
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Solution
The correct option is C
The given expansion can be written as
(1+x)(1+x)(1+x)........(1+x)n− factors
(1+y)(1+y).........(1+y)n− factors
(1+z)(1+z)(1+z).......(1+z)n− factors
There are 3n factors in this product.
To get a term of them of degree r, we choose r factors out of these 3n factors and then
multiply the second terms in each factor. There are 3nCr such terms each having coefficient 1.
Hence, the sum of the coefficients = 3nCr.
The given expansion can be written as
(1+x)(1+x)(1+x)........(1+x)n− factors
(1+y)(1+y).........(1+y)n− factors
(1+z)(1+z)(1+z).......(1+z)n− factors
There are 3n factors in this product.
To get a term of them of degree r, we choose r factors out of these 3n factors and then
multiply the second terms in each factor. There are 3nCr such terms each having coefficient 1.
Hence, the sum of the coefficients = 3nCr.