Perrmutations and Combinations
1. The multiplication principle or (r,s) principle
It states that if one operation can be performed in r ways, and a second operation can be performed in s ways, then the two operations can be performed in succession in r x s ways.
Note : The principle can be extended to any finite number of operations
2. Permutaion
(a). A permutaion is an ordered arrangement of all or part of a set of objects in a row.
(b). The umber of permutaion of n different objects taken all at a time (without repetition), is donated by nPn and nPn = n ! where n! = n(n – 1) (n – 2) ...x 3 x 2 x 1
(c). The number of permutations or arrangements of n different objects, taken r at a time (without repetition) is donated by nPr where 
(d). The number of permutations of n objects (not all distinc), taken all at a time, where thee are p objects alike of one kind, q objects alike of another kind and the rest are distinct, can be done in 
. (This result can be exdtended to many finite number of groups of objects).
3. Combination
(a). A combination is a selection of objects in which the order of selection does not matter.
(b). The number of combinations or selections of n different objects, taken r at the time, is donated by 
Note : nCo = 1 ; nCn = 1 ; nCr = nCn- r
(c). The number of selections from n different objects, taking any number at a time is 2n - 1
EXERCISES
1. A sixth from contains the Head Boy, the Head Girl and 8 other students. The form is asked to send a group of 4 representatives to a conference. Calculate the number of different ways i which the group can be formed if it must be contain :
(i). both the Head Boy and the Head Girl
(ii). Either the Head Boy or the Head Girl, but not both
2. Calculate the number of ways in which :
(i). 5 children can be devided in to group of 2 and 3
(ii). 9 children can be devided into groups of 5 and 4
Hence calculate the number of ways in which 9 children can be devided into groups of 2, 3, and 4
3. A row of 10 houses is to be painted in 3 colours, 2 houses ae to be red, 3 to be blue and 5 to be white. Find the number of different ways in which the row of the houses can be painted :
(i). with no restrictions
(ii). Given that the first and the last houses in the row are blue
(iii). Given that the first and the last houses in the row are the same colour.
4. Find the total numbers of different permutations of all the letters of the word RESERVE.
Find the number of these permutations in which :
(i). E is the first letter.
(ii). The two Rs come together
(iii). S and V come at the ends of the permutations.
5. Find the number of ways in which a team of 6 batsmen, 4 bowlers and a wicket-keeper may be selected from a squad of 8 batsmen, 6 bowlers and 2 wicket-keeper.
Find the number of ways in which :
(i). this team may be selected if it is to include 4 specified batsmen and 2 specified bowlers
(ii). The 6 batsmen may be selected from the 8 available, given that 2 particular batsmen cannot be selected together.
6. Calculate the total number of different permutations of all letters A, B, C, D, E and F when :
(i). there are no restriction
(ii0. the letters A and B are to be adjacent to one another
(iii). The first leter is A, B, or C and the last letter is D, E, or F.
7. A tennis team of 4 men and 4 women is to be picked from 6 men and 7 women. Find the number of ways in which this can be done.
It was decided that 2 of the 7 women must either be selected together or not selescted at all. Find how many possible teams could be selected in these circumstances.
The selected team is arranged into 4 pair, each consisting of a man an a woman.
Find the number of ways in which this can be done.
8. At an art exhibition 7 paintings are to be hung in a row along one wall. Find the number of possible arrangements
Given 3 paintings are by the same artist, find the number of arrangements in which :
(i). these 3 paintings are hung side by side.
(ii). Any one of these paintings in shung at the beginning of the row but neither of the other 2 is hung at the end of the row.
9. A shelf is to contain 7 different books, of which 4 were written by Dickens and 3 by Hardy. Find the number of aangements in which :
(i). no two books by the same author are adjacent
(ii0. the first two books at the left-hand end are by the same author.
10. A editor has space for 6 advertisements, one on each of the first 6 pages of the magazine. Of the 6 advertisements to be displayed, 4 ar for household goods, 1 for gardening equipment and 1 for sports equpipment. In how many different ways can the advertisements appear in the magazine if the 4 advertisements for household goods must appear on consecutive pages ?
Answers :
1. (i). 28 (ii). 112
2. (i). 10 126 ; 1 260
3. (i). 2 520 (ii). 168 (iii). 784
4. 420
(i). 180 (ii). 120 (iii). 20
5. 840, 72, 13
6. (i). 720 (i). 240 (iii). 216
7. 525, 225, 24
8. 5 040 (i). 720 (ii). 1 440
9. (i). 144 (ii). 2 160
10. 144
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