Practice section 5 of 10

A student wishes to re-arrange the letters and asterisks on the icon shown. Given that the letters of the word ACE must be together in the clockwise manner, find the number of possible arrangements if (i) the asterisks must be separated, (ii) the positions of the letters and asterisks are labelled.

A group of ten people consists of four single women, two single men and 2 couples. The ten people are seated randomly in a circle. Find the number of arrangements if (i) there is no restriction, (ii) the couples must seat together, (iii) the seats are of different colours.

Given the digits 1, 2, 3, 4, 6 and 8, and using each digit at most once, determine how many numbers can be formed if the number has to be

(i) less than 100,

(ii) between 100 and 1000,

(iii) between 1000 and 10000.

A girl wishes to phone a friend but cannot remember the exact number. She knows that it is a five-digit number, that it is even, and that it consists of the digits 2, 3, 4, 5 and 6 in some order. Using this information, find the largest number of different wrong telephone numbers she could try.

Find the number of ways in which 6 people may be divided into groups of

(i) 1, 2 and 3,

(ii) 2, 2 and 2,

(iii) 3, 1, 1 and 1.

A four-letter word can be formed from any of the 26 letters in the alphabet. Determine the number of such words with

(i) repetitions of the letters allowed,

(ii) different letters and containing exactly one vowel,

(iii) repetitions allowed and containing at least one vowel.