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# Combinations

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• ### Combinations

1. 1. Whats the Difference? "My fruit salad is a combination of apples,"The combination grapes andto the safe was bananas"472"
2. 2. • We dont care what order the fruits are in, they could also be "bananas, grapes and apples" or "grapes, apples and bananas", its the same fruit salad.• Now we do care about the order. "724" would not work, nor would "247". It has to be exactly 4-7-2.
3. 3. In Maths we use precise language..• If the order doesn’t matter, it is a COMBINATION• If the order does matter, it is a PERMUTATION
4. 4. In Maths we use precise language.. • If the order doesn’t matter, it is a COMBINATION • If the order does matter, it is a PERMUTATIONA PERMUTATION IS AN ORDERED COMBINATION.
5. 5. n The ( ) Notation r n• Can also be written as C aswell as nCr r and C(n,r).• It gives the number of ways of choosing r objects from n different objects.• It is pronounced ‘n-c-r’ or ‘n-choose-r’.
6. 6. How to Calculate It. n) =(r n! n) = n(n - 1)(n - 2)...(n - r +1) (r r! (n - r)! r!
7. 7. How to Calculate It. n) =(r n! n) = n(n - 1)(n - 2)...(n - r +1) (r r! (n - r)! r! Deﬁnition!
8. 8. How to Calculate It. n) =(r n! n) = n(n - 1)(n - 2)...(n - r +1) (r r! (n - r)! r! Deﬁnition! Practical!
9. 9. You have a go!n0nn
10. 10. You have a go!• Question 3 on your worksheet. n 0 n n
11. 11. You have a go!• Question 3 on your worksheet.• Answer 15. n 0 n n
12. 12. You have a go!• Question 3 on your worksheet.• Answer 15.• And Question 4. n 0 n n
13. 13. You have a go!• Question 3 on your worksheet.• Answer 15.• And Question 4.• (a) (n) = 1 0 n n
14. 14. You have a go!• Question 3 on your worksheet.• Answer 15.• And Question 4.• (a) (n) = 1 0• (b) (n) = 1 n
15. 15. Now a twist• Assume you have 13 soccer players and you can pick only 11 to play.• How many ways can you choose those players - Question 5.
16. 16. • You can also ﬁnd it this way!• Think of it .. every time you choose 11 you don’t choose 2!• 13) = (13) = 13 × 12 = 78 Thus (12 2 2×1
17. 17. The Twin Rule n) = ( n)• It states that ( r n-r• Proof:LHS = =RHS = = n! = n! = LHS (n - r)!(n - (n - r))! (n - r)!r!
18. 18. Equations using (n-c-r) When you have to solve equations the following are very usefull. n n(1) = 1 ( 2 ) = n(n - 1) = n(n - 1) 2×1 2
19. 19. Example• Solve for the value of the natural number n n such that ( ) = 28. 2
20. 20. Solutionn(n - 1) = 28 2 n^2 - n = 28 2 n^2 - n = 28 -> n^2 - n - 28 = 0 (n - 8)(n + 7) = 0 n=8 n=-7 Reject n = - 7 is not a natural number. Therefore n = 8.