Permutations and Combinations Quiz Set 008

This word has 7 letters, out of which 4 are consonants and 3 are vowels. The vowels have to occupy three contiguous positions. This triad can be arranged in 3! ways like this: we can place 3 vowels in first place, 2 in second place and 1 in the third place, giving 3 × 2 × 1 = 6 permutations. Next, we have to arrange the 4 consonants and the triad treated as one letter, giving 5! = 120 possibilities. So the total possibilities are 6 × 120 = 720.

The secretary can be any of the 19 members, the general-secretary can then be any of the (19 - 1) members. So the answer is 19 × (19 - 1) = 342.

This word has 10 different letters. The number of possibilities is ¹⁰P₄ = 5040.

The first can get any of the 9 ice-creams, the second can then get (9 - 1), and so on. This is expressed as ⁹P₅, which evaluates to 15120.

This word has 6 letters, out of which 3 are consonants and 3 are vowels. The vowels have to occupy three fixed odd positions. We can place 3 vowels in first odd place, 2 in second odd place and 1 in the third odd place, giving 3 × 2 × 1 = 6 permutations. This will be done with the consonants also. So the total possibilities are 6 × 6 = 36.