Prove that in the expansion of `(1+x)^(n) (1+y)^(n) (1+z)^(n)`, the sum of the coefficient of the terms of degree r is `.^(3n)C_(r )`.

Options

A. `(""^(n)C_(r))^(3)`
B. `3 . ""^(n)C_(r)`
C. `""^(3n)C_(r)`
D. `""^(n)C_(3r)`

Answer

c

Explanation

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