Question

Column – 1 : represent different words
Column – 2 : represent number ways of selecting five letters from the word in column – 1
Column – 3 : represent total number of possible words with or without meaning, using all the alphabets of word in such that all the vowels are together.
$\begin{array}{ccc}Column1& Column2& Column3\\ \left(I\right)INDEPENDENT& \left(i\right)41& \left(p\right){24.}^8{C}_6{.}^6{C}_3{.}^3{C}_2{.}^3{C}_2\\ \left(II\right)INSTITUTE& \left(ii\right)60& \left(Q\right){24.}^8{C}_5{.}^5{C}_3\\ \left(III\right)CURRICULUM& \left(iii\right)72& \left(R\right){12.}^7{C}_3{.}^4{C}_3{.}^3{C}_2\\ \left(IV\right)MATHEMATICS& \left(iv\right)179& \left(S\right){24.}^6{C}_3{.}^3{C}_2\end{array}$
Which of the following options is the only CORRECT combination?

• A. (I) (i) (Q)
• B. (IV) (iii) (P)
• C. (II) (iv) (S)
• D. (III) (ii) (R)

Answer 1

The correct option is D (III) (ii) (R)

$\begin{array}{ccccc}& INDEPENDENT& CURRICULUM& INSTITUTE& MATHEMATICS\\ Choice& Ways& Ways& Ways& Ways\\ Alldistinct& {}^6{C}_5=6& {}^6{C}_5=6& {}^6{C}_5=6& {}^6{C}_5=6\\ 3distinct,2alike& {}^3{C}_1{\times }^5{C}_3=30& {}^3{C}_1{\times }^5{C}_3=30& {}^2{C}_1{\times }^5{C}_3=20& {}^3{C}_1{\times }^7{C}_3=105\\ 2distinct,3alike& {}^2{C}_1{\times }^5{C}_2=20& 1{\times }^5{C}_2=10& 1{\times }^5{C}_2=10& \cdots \cdots \\ 2alike,2alike,1distinct& {}^3{C}_2{\times }^4{C}_1=12& {}^3{C}_2{\times }^3{C}_1=12& 1\times 4=4& {}^3{C}_2{\times }^6{C}_1=18\\ 3alike,2alike& {}^2{C}_1{\times }^2{C}_1=4& 1{\times }^2{C}_1=2& 1\times =1\times \cdots \cdots & \\ Total& 72& 60& 41& 179\end{array}$
Number of words such that all vowels are together
$INSTITUTE=\frac{6!}{3!}\times \frac{4!}{2!}={24.}^6{C}_3{.}^3{C}_2$
$CURRICULUM=\frac{7!}{2!2!}\times \frac{4!}{3!}={12.}^7{C}_3{.}^4{C}_3{.}^3{C}_2$
$MATHEMATICS=\frac{8!}{2!2!}\times \frac{4!}{2!}={24.}^8{C}_6{.}^6{C}_3{.}^3{C}_2{.}^3{C}_2$
So, Match the answers accordingly Ans.49-(D), 50-(C), 51-(D)