Find the coefficient of ` a^(3) b^(4)c^(5) ` in the
expasion of `(bc + ca + ab)^(6)`

Explanation

In this case , write ` a^(5) b^(4) c^(5) = (ab)^(x) (bc)^(y) (ca)^(x) ` say
` therefore a^(3) b^(4) c^(5) = a^(z + x) . b^(x+y) . c^(y+ x)`
` rArr z + x = 3, x + y = 4 `
y + z = 5
On adding all , we get `2(x + y + z) = 12`
` therefore x + y + z = 6`
Then x = 1, x = 3, x = 2
Therefore , the coefficient of `a^(3)b^(4)c^(5)` in the expansion of
`(bc + ca + ab)^(6)` or the coefficient of `(ab)^(1) (ca)^(5) (ca)^(2)` in the
expansion of `(bc + ca + ab)^(4)"is"(6!)/(1!3!2!) , i.e., 60`.
Aliter
coefficient of `a^(3)b^(4) c^(5)` in the expansion of `(bc + ca + ab)^(6)`
= Coefficient of `(abc)^(6) ((1)/(a) + (1)/(b) + (1)/(c))^(6)`
= Coefficient of `((1)/(a))^(3) ((1)/(b))^(2) ((1)/(c))^(1)` in the expansion of
` ((1)/(a) + (1)/(b) + (1)/(c))^(6) "is" (6!)/(3!2!1!) = 60`