Permutations and Combinations

How many different number plates, passwords or teams can be formed from a given set of objects? This chapter develops systematic methods of counting. It begins with the fundamental principle of counting: if one event can occur in m ways and a second event in n ways, both can occur in m x n ways. The factorial notation n! = 1 x 2 x 3 x ... x n is introduced, with 0! = 1. A permutation is an arrangement of objects in a definite order, and the number of permutations of n different objects taken r at a time is nPr = n!/(n - r)!. You will also study permutations when some objects are alike, given by n!/(p1! p2! ... pk!), and arrangements with repetition. A combination is a selection in which order does not matter, and nCr = n!/(r!(n - r)!). Important results include nCr = nC(n - r) and nCr + nC(r - 1) = (n + 1)Cr. These ideas are applied to forming words and numbers, choosing committees and selecting cards.

Detailed Worksheet: Permutations and Combinations Section A - Definitions (10 marks) 1. State the fundamental principle of counting with an example. (2 marks) 2. Evaluate 8!/(6! x 2!). (2 marks) 3. Find the number of 3-digit numbers that can be formed from the digits 1, 2, 3, 4 and 5 if repetition is allowed. (2 marks) 4. If nC8 = nC2, find n and hence nC2. (2 marks) 5. Distinguish between a permutation and a combination. (2 marks) Section B - Calculations and Applications (15 marks) 6. How many words, with or without meaning, can be formed from the letters of the word MISSISSIPPI? (3 marks) 7. In how many ways can a committee of 3 men and 2 women be chosen from 5 men and 4 women? (3 marks) 8. How many even 3-digit numbers can be formed from the digits 1, 2, 3, 4, 6 and 7 if repetition is allowed? (3 marks) 9. If nPr = 840 and nCr = 35, find r and n. (3 marks) 10. Find the number of diagonals of a hexagon. (3 marks) Section C - Diagrams (10 marks) 11. Draw a tree diagram to show the outfits possible from 3 shirts and 2 pairs of trousers, and state the total. (4 marks) 12. Draw a box diagram showing how the vowels of EQUATION are kept together as one unit, and find the number of such arrangements. (3 marks) 13. Draw a slot diagram for the number of 5-digit telephone numbers that start with 67 and have no repeated digits. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. In how many ways can the letters of the word PERMUTATIONS be arranged if (i) the words start with P and end with S (ii) the vowels are all together? (5 marks) 15. A cricket team of 11 players is to be chosen from 17 players, of whom 5 are bowlers. In how many ways can it be chosen if it must include exactly 4 bowlers? (5 marks) 16. How many words, with or without meaning, each of 2 vowels and 3 consonants, can be formed from the letters of the word DAUGHTER? (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Show the formula used and all working.