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Find the coefficient of a^3b^4c^5 in the...

Find the coefficient of `a^3b^4c^5` in the expansion of `(b c+c a+a b)^6dot`

Text Solution

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In this case, write `a^(3)b^(4)c^(5) = (ab)^(n_(1))(bc)^(n_(2))(ca)^(n_(3))` (say)
`:. a^(3)b^(4)c^(5) = (a)^(n_(1)+n_(3))(b)^(n_(1)+n_(2))(c )^(n_(2)+n_(3))`
`rArr n_(1) + n_(3) = 3, n_(1) + n_(2) = 4, n_(2) + n_(3) = 5`
Adding, we get
`2(n_(1)+n_(2)+n_(3)) = 12`
`rArr n_(1)+n_(2)+n_(3) =6`
So, `n_(1) = 1, n_(2) = 3, n_(3) = 2`
Therefore, the required coefficient is `(6!)/(1! xx 3! xx 2!) = 60`
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