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5.7 The Binomial Theorem

5.7 The Binomial Theorem. Formulas for C ( n,r ) Binominal Coefficient Binomial Theorem Number of Subsets. Formulas for C ( n,r ). Example Routes. Work the route problem covered previously by selecting where in the string of length 7 the 4 E’s will be placed instead of the 3 S’s.

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5.7 The Binomial Theorem

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  1. 5.7 The Binomial Theorem • Formulas for C(n,r) • Binominal Coefficient • Binomial Theorem • Number of Subsets

  2. Formulas for C(n,r)

  3. Example Routes • Work the route problem covered previously by selecting where in the string of length 7 the 4 E’s will be placed instead of the 3 S’s. • Therefore the total number of possible routes is Notice that C(7,4) = C (7,3).

  4. Binominal Coefficient • Another notation for C(n,r) is . is called a binominal coefficient.

  5. Example Binominal Coefficient

  6. Binomial Theorem • Binomial Theorem

  7. Example Binominal Theorem • Expand (x + y )5. (x + y )5 = x5 + 5x4y + 10x3y2 + 10x2y3+ 5xy4 + y5

  8. Number of Subsets • A set with n elements has 2n subsets.

  9. Example Number of Subsets • A pizza parlor offers a plain cheese pizza to which any number of six possible toppings can be added. How many different pizzas can be ordered? • Ordering a pizza requires selecting a subset of the 6 possible toppings. • There are 26 = 64 different pizzas.

  10. Summary Section 5.7 - Part 1 • C(n,r) is also denoted by . • The formulaC(n,r) = C(n,n - r) simplifies the computation of C(n,r) when r is greater than n/2. • The binomial theorem states that

  11. Summary Section 5.7 - Part 2 • A set with n elements has 2n subsets.

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