Question

$\large\mathcal{\colorbox{gold}{\color{red}{Question~:}}}$
$\tt{The~coefficent~of~5th~term~in~the~expression~of$ $\dfrak{x}{ - 3y)}^{7}$
1) 105
2) 104
3) 103
4) 110​

Answer 1

$\huge\fbox\red{A}\fbox\orange{n}\fbox\purple{s}\fbox\green{w}\fbox\pink{e}\fbox\blue{r}$

$\huge\red{use \: binomial\: theoram }$

$[math](x+y+1)^{12}=\displaystyle \sum_{k=0}^{12}\binom{12}{k}x^{12-k}(y+1)^k[/math]$

Now select the term with 12-k=3, which is k=9. For this k , the coefficient with all the y’s is $[math](y+1)^9=\displaystyle \sum_{i=0}^{k (=9)} \binom{k}{i} y^i[/math]$

Now choose from this sum the coefficient in front of $[math]y^7 [/math], which is [math]\binom{9}{7}[/math]$

There you have it, the coefficient in front of $[math]x^3 y^7[/math] is [math]\binom{12}{9} \binom{9}{7}=7920[/math]$