**Question 1 :**A family of 4 brothers and 3 sisters is to be arranged in a row for a photograph. In how many ways can they be seated if all the sisters are together?

**Answer:**Let B

_{1},B

_{2},B

_{3},B

_{4}denote the brothers and S

_{1},S

_{2},S

_{3}denote the sisters. Since the sisters are to be seated together for a photograph, consider all the sisters as one unit or entity. Then B

_{1},B

_{2},B

_{3},B

_{4},S can be arranged to sit in 5! ways. The sisters can be arranged among themselves in 3! ways. Since the two events are independent, the total number of arrangements = 5!.3! = 720 ways.

**Question 2:**In how many ways can a consonant and a vowel be chosen out of the letters in the word COURAGE?

**Answer:**There are three consonants (C, R, G) and four vowels (A, E, O, U) in the word COURAGE.

With the consonants, we may choose any one of the 4 vowels. It can be done in 4 ways. There are three consonants.

\The total number of ways will be 4 x 3 = 12.

**Question 3:**How many arrangements can be made out of the letters of the word DRAUGHT, the vowels never being separated?

**Answer:**There are 7 letters in the word DRAUGHT, the two vowels are A and U. Since, the vowels are not to be separated, AU can be considered as one entity. Therefore, the number of letters will be 6 instead of 7. The permutations will be P(6,6) = 6! ways.

But the two vowels A and U can be arranged in two ways, i.e. AU and UA.

\The required number of arrangements = 2!.6! = 1440 ways.

**Question 4:( Out of SPM Syllibus)**Find the number of arrangements that can be made out of the letters i) ASSASSINATION ii) GANESHPURI.

**Answer:**i) The word ASSASSINATION consists of

A's = 3, S's = 4, I's = 2,

N's = 2, T's = 1, O's = 1

The total number of letters is 13 letters.

ii) The word GANESHPURI consists of 10 distinct letters.

The number of permutations is 10!.

**Question 5 :**( out of SPM Syllibus) How many different arrangements can be made out of the letters in the expression a

^{3}b

^{2}c

^{4}, when written at full length?

**Answer:**There are 3 + 2 + 4 = 9 letters.

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