Combinations

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Combinations

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

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