The coefficient ofa a3 b4 c5 in b c-c a-a b6 is

The correct option is A 60 [If n is a positive integer and a_1a_2a_3........a_m ∈ C then (a_1+a_2+a_3+.....a_m)^n=∑n!/n_1!n_2!.......n_m!a_1^n_1 a_2^n_2 a_3^ ...

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Byju's Answer

Standard XII

Mathematics

Range of Quadratic Expression

The coefficie...

Question

The coefficient of $a^{3} b^{4} c^{5}$ in $(b c + c a + a b)^{6}$ is

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Solution

The correct option is A
60

$[$ If n is a positive integer and $a_{1} a_{2} a_{3} . . . . . . . . a_{m} \in C$ then
$(a_{1} + a_{2} + a_{3} + . . . . . a_{m})^{n} = \sum \frac{n!}{n_{1}! n_{2}! . . . . . . . n_{m}!} {a_{1}}_{1}^{n} {a_{2}}_{2}^{n} {a_{3}}_{3}^{n} . . . . . {a_{m}}_{m}^{n}$ where $n_{1} n_{2} n_{3} . . . . n_{m}$ are all non-negative integers subject to the condition $n_{1} + n_{2} + n_{3} + . . . . . + n_{m} = n]$
$a^{3} b^{4} c^{5} = (a b)^{x} (b c)^{y} (c a)^{z}$
= $a^{z + x} b^{x + y} c^{y + z}$
$\Rightarrow z + x = 3, x + y = 4, y + z = 5 \Rightarrow x + y + z = 6$
then x=1, y=3, z=2
The co-efficient of $a^{3} b^{4} c^{5} = \frac{6!}{1! 2! 3!} = 60$

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The coefficient ofa3b4c5 in (bc+ca+ab)6 is

The coefficient of $a^{3} b^{4} c^{5}$ in $(b c + c a + a b)^{6}$ is