Permutation And Combination - Aptitude MCQ Questions and Solutions with Explanations

The given word contains 7 letters out of which E is taken 2 times and all other letters are different .
∴ Required number of ways
$$\eqalign{ & = \frac{{7!}}{{2!}} \cr & = \frac{{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}}{2} \cr & = 2520 \cr} $$

Required number of ways
$$\left( {{}^5{{\text{C}}_1} \times {}^4{{\text{C}}_2}} \right) + \left( {{}^5{{\text{C}}_2} \times {}^4{{\text{C}}_1}} \right)$$     $$ + \left( {{}^5{{\text{C}}_3}} \right)$$
$$ = \left( {5 \times \frac{{4 \times 3}}{{2 \times 1}}} \right)$$   $$ + \left( {\frac{{5 \times 4}}{{2 \times 1}} \times 4} \right)$$   $$ + \left( {\frac{{5 \times 4 \times 3}}{{3 \times 2 \times 1}}} \right)$$
$$\eqalign{ & = \left( {30 + 40 + 10} \right) \cr & = 80 \cr} $$

Required number of ways
$$\eqalign{ & \left( {{}^8{C_5} \times {}^{10}{C_6}} \right) + \left( {{}^8{C_3} \times {}^{10}{C_4}} \right) \cr & = \frac{{8 \times 7 \times 6}}{{3!}} \times \frac{{10 \times 9 \times 8 \times 7}}{{4!}} \cr & = \frac{{8 \times 7 \times 6}}{{3 \times 2 \times 1}} \times \frac{{10 \times 9 \times 8 \times 7}}{{4 \times 3 \times 2 \times 1}} \cr & = 11760 \cr} $$

The given 6 letters, all different.
∴ Required number of ways
$$\eqalign{ & {}^6{P_6} = 6! \cr & = {6 \times 5 \times 4 \times 3 \times 2 \times 1} \cr & = 720 \cr} $$

The given word contains 6 letter out of which R is taken 2 times, U is taken to 2 times and other letters are all different.
∴ Required number of ways
$$\eqalign{ & = \frac{{6!}}{{2! \times 2!}} \cr & = \frac{{6 \times 5 \times 4 \times 3 \times 2 \times 1}}{{2 \times 2}} \cr & = 180 \cr} $$

The word ‘TRANSPIRATION’ has 13 letters in which each of T, R, A, N and I has come two times
We have to arrange TT RR NN PS (AA II O)
There are five vowels in the given words.
∴ We consider these give vowels as one letter.
∴ Required number of arrangements
$$\eqalign{ & = \frac{{9! \times 5!}}{{2!\, 2! \,2! \,2! \,2!}} \cr & = \frac{{9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 5 \times 4 \times 3 \times 2}}{{2 \times 2 \times 2 \times 2 \times 2}} \cr & = 1360800 \cr} $$

Keeping the vowels (AIA) together, we have CPTL (AIA).
We treat (AIA) as 1 letter.
Thus, we have to arrange 5 letters.
These can be arranged in 5! = (5 × 4 × 3 × 2 × 1) ways = 120 ways
Now, (AIA) are 3 letters with 2A and 1I
These can be arranged among themselves in
$$\frac{{3!}}{{2!}} = \frac{{3 \times 2 \times 1}}{{2 \times 1}} = 3$$     ways
∴ Required number of ways = 120 × 3 = 360

Required number of ways
$$\eqalign{ & = {{}^4{C_2} \times {}^5{C_3}} \cr & = {{}^4{C_2} \times {}^5{C_2}} \cr & = {\frac{{4 \times 3}}{{2 \times 1}} \times \frac{{5 \times 4}}{{2 \times 1}}} \cr & = 60 \cr} $$

The given words contains 8 letters of which E is taken 2 times.
∴ Required number of ways
$$\eqalign{ & = \frac{{8!}}{{2!}} \cr & = \frac{{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}}{2} \cr & = 20160 \cr} $$

The given words contains 6 letters of which E is taken 2 times.
∴ Required number of ways
$$\eqalign{ & = \frac{{6!}}{{2!}} \cr & = \frac{{6 \times 5 \times 4 \times 3 \times 2!}}{{2!}} \cr & = 360 \cr} $$