Mathematics - Permutation & Combination Question with Solution | TestHub

A-Q: Coefficient of $x^{18}$ in ${\left(1+x+\cdots +x^{12}\right)}^3$ $\begin{array}{rlr}& ={\left(1-x^{13}\right)}^3(1-x{)}^{-3}& =\left(1-3x^{13}\right)(1+{}^3{C}_{1}x+\left({}^4{C}_2\right)x^2+\cdots \cdots \end{array}$

Which is ${}^{20}{C}_{18}-3^7{C}_5={}^{20}{C}_2-3^7{C}_2=190-63=127$

B-P : A team of 5 can be prepared in having 3 G \& 2 B and 4 G \& 1 B Total ways $={}^5{C}_3\times {}^5{C}_2+{}^5{C}_4\times {}^5{C}_1=125$

C-S : We first fill the small boxes with 2 of the three bundles and the remaining boxes with the remaining bundles. The desired number is ${}^3{P}_2\cdot 4!=6\cdot 24=144$

D-R : We choose any colour and take 2 balls or take any number from 1-6 and choose 2 colours. The desired number is ${}^5{C}_1{}^6{C}_2+{}^6{C}_1{}^5{C}_2=75+60=135$