Binomial Expansion and Coefficients

Out of the 5 factors, 1 of them is chosen to be an $a$ : ${\large\binom{5}{1}}a$

Out of the remaining 4 factors, 2 of them are chosen to be $2b$ : ${\large\binom{4}{2}}(2b)^2$

Out of the remaining 2 factors, 2 of them are chosen to be $3c$ : ${\large\binom{2}{2}}(3c)^2$

${\large\binom{5}{1}}a\times{\large\binom{4}{2}}(2b)^2\times{\large\binom{2}{2}}(3c)^2={\large\binom{5}{1}\binom{4}{2}\binom{2}{2}}(36)ab^2c^2$

$=(5)(6)(1)(36)ab^2c^2=1080ab^2c^2$

The coefficient of the term is $\boxed{1080}$ .

-( a + 2b + 3c )^5 = (( a + 2b ) + 3c )^5 = ( a + 2b )^5 + 5( a + 2b )^4.( 3c ) + 10( a + 2b )^3. (3c)^2 + 10( a + 2b )^2. (3c)^3 + 5( a + 2b )(3c)^4 + (3c)^5 - Consider 10( a+ 2b )^3. ( 3c )^2 = 90c^2( a^3 + 6a^2b + 12ab^2 + 8b^3 ) = 90a^3c^2 + 540a^2bc^2 + 1080ab^2c^2 + 720b^3c^2

So, the coefficient of ab^2c^2 = 1080