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Question: Find the coefficient of \(a^{3}b^{4}c^{5}\) in the expansion of \((bc + ca + ab)^{6}\)...

Find the coefficient of a3b4c5a^{3}b^{4}c^{5} in the expansion of (bc+ca+ab)6(bc + ca + ab)^{6}

A

0

B

60

C

– 60

D

None of these

Answer

60

Explanation

Solution

In this case, a3b4c5=(ab)x(bc)y(ca)z=ax+z.bx+y.cy+za^{3}b^{4}c^{5} = (ab)^{x}(bc)^{y}(ca)^{z} = a^{x + z}.b^{x + y}.c^{y + z}

z + x = 3, x+y=4,y+z=5x + y = 4,y + z = 5; 2(x+y+z)=122(x + y + z) = 12; x+y+z=6x + y + z = 6. Then x=1,y=3,z=2x = 1,y = 3,z = 2

Therefore the coefficient of a3b4c5a^{3}b^{4}c^{5} in the expansion of (bc+ca+ab)6(bc + ca + ab)^{6} = 6!1!3!2!=60\frac{6!}{1!3!2!} = 60.