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The coefficients in the binomial expansion of (1 + x ) 5 .

Want to know some BRAIN BLOWING MATHS?. 1 5 10 10 5 1. The coefficients in the binomial expansion of (1 + x ) 5. The coefficient of x 6 in the expansion of (1 + x ) 49 is 49 C 6 , That’s the number of ways of winning the jackpot on the National Lottery.

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The coefficients in the binomial expansion of (1 + x ) 5 .

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  1. Want to know some BRAIN BLOWING MATHS? 1 5 10 10 5 1 The coefficients in the binomial expansion of (1 + x) 5. The coefficient of x 6 in the expansion of (1 + x) 49 is 49 C 6 , That’s the number of ways of winning the jackpot on the National Lottery.

  2. The number of ways of winning the jackpot on the National Lottery is 13 983 816 13 983 816 two pence pieces laid end to end would stretch 220 miles – from London to Paris. ……………… ……………………. 13 983 816 seconds is 161 days – from 11th April until 19th September.

  3. INTRODUCTION The Binomial ExpansionFor Positive Integers Lets start at the very beginning! Mrs Richards www.mathsathawthorn.pbwiki.com

  4. STARTER 1. What is the value of two to the power of four ? 2. What is the cube of 2x ? NOT 3. What is the square of 5y ? 5. What is the cube of 4. Simplify 6. What is any number to the power of zero ? 7. What is ( a + b ) to the power of zero ? That’s all we can do 8. Expand and simplify 9. Expand and simplify Mrs Richards

  5. ANOTHER WAY….. ONE WAY….. Mrs Richards

  6. We know that: Observations? POWERS TOTAL 2 in each TERM KEY WORDS TERM COEFFICIENT We call the expansion Binomial as the original expression has 2 parts. Mrs Richards

  7. 1 2 1 Mrs Richards

  8. Expand these Binomials: 1. (a+b)0 2. (a+b)1 3. (a+b)2 4. (a+b)3 5. (a+b)4 Mrs Richards

  9. 1 1 3 3 Collecting like terms gives: Observations? We again see that there is a clear pattern in the TERMS. SYMMETRY POWERS TOTAL 3 in each TERM Mrs Richards

  10. 1 1 3 3 Mrs Richards

  11. Collecting like terms gives: Observations? SYMMETRY POWERS TOTAL 4 in each TERM Mrs Richards

  12. Mrs Richards

  13. Spot anything? • 1. (a+b)0 • 2. (a+b)1 • 3. (a+b)2 • (a+b)3 • (a+b)4 =1 =1a+1b =1a2+2ab+1b2 =1a3+3a2b+3ab2+1b3 =1a4+4a3b+6a2b2+4ab3+1b4 Mrs Richards

  14. Powers of a + b Pascal’s Triangle Expression 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 The numbers of each term in the expansion match PASCAL’s TRIANGLE Mrs Richards

  15. PREDICTION? 1, 5, 10, 10, 5, 1 1, 6, 15, 20, 15, 6, 1 Mrs Richards

  16. 1 1 3 3 How is this useful? EXPAND 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 Power of 0 Power of 1 Power of 2 Power of 3 here a is replaced with x and b is replaced with 2 Power of 4 Mrs Richards

  17. Slightly harder? EXPAND 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 Power of 0 Power of 1 Power of 2 Power of 3 Power of 4 RUBBISH! Mrs Richards

  18. 1. WJEC January 2006 C1 PAST PAPER QUESTION We start with PASCAL’s bis replaced with (2) Here a is replaced with (3x) The use of brackets will help avoid errors . Mrs Richards

  19. Try for yourself (3+2x)4 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 Power of 0 Power of 1 Power of 2 Power of 3 Power of 4 Mrs Richards

  20. 2. WJEC May 2011 C1 PAST PAPER QUESTION Here a is replaced with 3 bis replaced with (2x) The importance of the brackets must be stressed Mrs Richards

  21. 3. WJEC Jan 2010 C1 PAST PAPER QUESTION Jan 2010 We KNOW n=5 and so we can use PASCALS TRIANGLE if we wish. TERM in x TERM in x squared COEFFICIENT of the TERM in x COEFFICIENT of the TERM in x squared Mrs Richards

  22. TERMS COEFFICIENTS Solve this equation….. This is a taste of future stories! Mrs Richards

  23. 3. WJEC January 2012 C1 Introducing Fractions Here a is replaced with x bis replaced with 2 2 Mrs Richards

  24. 4. WJEC May 2009 C1 Mrs Richards

  25. 5. WJEC JANUARY 2009 C1 Here a is replaced with 1/4 bis replaced with (2x) BUT! We are not interested in all terms. We must think carefully, which TERM will involve x cubed? This is the TERM that will involve x cubed Mrs Richards

  26. 5. WJEC JANUARY 2009 C1 Here a is replaced with 1/4 bis replaced with (2x) BUT! We are not interested in all terms. We must think carefully, which TERM will involve x cubed? The TERM containing x cubed is: Which simplifies to: This is the TERM that will involve x cubed 2 5 So the COEFFICIENT of x cubed is Mrs Richards

  27. WJEC May 2012 C1 PAST PAPER QUESTION Using PASCAL’S TRIANGLE First consider the expansion of Here a is replaced with 1 bis replaced with (-2x) Mrs Richards

  28. WJEC May 2007 C1 PAST PAPER QUESTION Mrs Richards

  29. WJEC January 2013 C1 Mrs Richards

  30. WJEC January 2005 C1 Mrs Richards

  31. WJEC January 2007 C1 But we are told this equals… (given in question) So solve the equation Mrs Richards

  32. MAY 2010 Mrs Richards

  33. WJEC January 2011 C1 Mrs Richards

  34. Mrs Richards

  35. The Story Continues….. Mrs Richards

  36. PASCAL DEFINED THE NUMBER OF WAYS OF SELECTING rOBJECTS FROM n OBJECTS AS: Sometimes written as Mrs Richards

  37. June 2009 part b Is the TERM in x squared Is the COEFFICIENT of x squared SOLVE this Mrs Richards

  38. Coefficient of x squared is twice the coefficient of x So n=3 Mrs Richards

  39. JUNE 2007 part b Mrs Richards

  40. JAN 2012 part b Mrs Richards

  41. Mrs Richards

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