Find the totoal number of distnct or
dissimilar terms in the expansion of
` (x + y + z + w)^(n), n in N `

Explanation

The total number of distinct or dissimilar terms in the
expasion of `(x + y + z + w)^(n)` is
`= ""^(n+ 4-1)C_(4) = ""^(n+3)C_(3) = ((n+3)(n+2)(n+1))/(1*2*3)`
` = ((n+1)(n+2)(n+3))/(6)`
I . Aliter
We know that , `(x + y + z+ w)^(n) = {(x + y) + (z)}^(n)`
`+ ""^(n)C_(2) (x + y)^(n-2) (z + w)`
` + ""^(n)C_(2) (x+y)^(n-2) (z + w)^(2) + ...+ "^(n)C_(n) (z + w)^(n)`
` therefore ` Number of terms is RHS
` = (n+1)+ n.*2+ (n-1).3 ...+ .(n+1)`
`sum_(r=0)^(n) (n-r+1)(r+1)`
`= sum_(r-0)^(n) (n+1)+ nr -r^(2)= (n+1)sum_(r=0)1+n sum_(r=0)^(n) r - sum_(r=0)^(n) r^(2)`
`(n+1)*(n+1) + n. (n(n+1))/(2) - (n(n+1)(2n =1))/(6)`
` = ((n+1)(n+2)(n+3))/(6)`
II. Aliter
`(x+ y+ z + w)^(n)= sum (n!)/(n_(1)!n_(2)!n_(3)!n_(4)) x^(n_(1)) y^(n_(2)) z^(n_(3))w^(n_(4))`
where , `n_(1),n_(2),n_(3),n_(4) = n`
Hence , number of the distinct terms
= Coefficient of `x^(n) "in "(x^(0) + x^(1) + x^(2) + ...+ x^(n))^(4)`
= Coefficient of `x^(n) "in "((1 - x^(x+1))/(1-x))^(4) `
= Coefficient of `x^(n) "in " (1 - x^(n+1))^(4)(1-x)^(-4)`
= Coefficient of `x^(n) "in "(1-x)^(-4) " " [ because x^(n+1) gt x^(n)]`
`=""^(n+3)C_(n) = ""^(n+3)C_(3) = ((n+3)(n+2)(n+1))/(6)`