Test: Permutation & Combination- 2 - UPSC MCQ

The total number of ways to select 4 children out of 10 total children is (104)=10∗9∗8∗74∗3∗2∗1=210. However, because at least one boy must be in the group, we have to subtract 1 to account for the case where all four children are girls (There is (44)=1 such case). The answer is 210−1=209.

The word 'OPTICAL' contains 7 different letters. When the vowels OIA are always together, they can be supposed to form one letter. Then, we have to arrange the letters PTCL (OIA). Now, 5 letters can be arranged in 5!=120 ways. The vowels (OIA) can be arranged among themselves in 3!=6 ways. Required number of ways =(120∗6)=720.

Vowels in the word "CORPORATION" are O,O,A,I,O
Lets make it as CRPRTN(OOAIO)
This has 7 lettes, where R is twice so value = 7!/2!
= 2520
Vowel O is 3 times, so vowels can be arranged = 5!/3!
= 20
Total number of words = 2520 * 20 = 50400

5 men can be chosen out of 7 men in ⁷C₅ = ⁷C₂ = 21 ways.

2 women out of 3 women in ³C₂ = ³C₁ = 3 ways.

Hence by fundamental counting principle, group of 5 men and 2 women can be chosen from 7 men and 3 women in (21)(3) = 63 ways

In the word 'MATHEMATICS', we'll consider all the vowels AEAI together as one letter.
Thus, we have MTHMTCS (AEAI).
Now, we have to arrange 8 letters, out of which M occurs twice, T occurs twice
 Number of ways of arranging these letters =8! / ((2!)(2!))= 10080.

Now, AEAI has 4 letters in which A occurs 2 times and the rest are different.
Number of ways of arranging these letters =4! / 2!= 12.

 Required number of words = (10080 x 12) = 120960

Required number of ways= (⁸C₅ x ¹⁰C₆)
= (⁸C₃ x ¹⁰C₄)
= (8 x 7 x 6 /3 x 2 x 1)  x  (10 x 9 x 8 x 7 / 4 x 3 x 2 x 1)
= 11760.

The word 'LOGARITHMS' has 10 different letters.
Hence, the number of 3-letter words(with or without meaning) formed by using these letters 
= ¹⁰P₃
= 10 x 9 x 8
= 720

The word 'LEADING' has 7 different letters.

When the vowels EAI are always together, they can be supposed to form one letter.

Then, we have to arrange the letters LNDG (EAI).

Now, 5 (4 + 1) letters can be arranged in 5! = 120 ways.

The vowels (EAI) can be arranged among themselves in 3! = 6 ways.

Required number of ways = (120 x 6) = 720.