By Alan Doerr, Kenneth Levasseur

Textbook from UMass Lowell, model 3.0

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Applied Discrete constructions by means of Alan Doerr & Kenneth Levasseur is authorized below an inventive Commons Attribution-NonCommercial-ShareAlike 3.0 usa License.

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We can rewrite the expansion as ✓ ◆ ✓ ◆ ✓ ◆ ✓ ◆ ✓ ◆ ✓ ◆ 5 5 5 4 5 3 2 5 2 3 5 5 5 4 x + x y+ x y + x y + xy + y 0 1 2 3 4 5 In summary, in the expansion of (x + y)n we note: 1. The first term is xn and the last term is y n . 4. COMBINATIONS AND THE BINOMIAL THEOREM 39 2. With each successive term, exponents of x decrease by 1 as those of y increase by 1. For any term the sum of the exponents is n. 3. The coeﬃcient of xn y is k k n k . 4. The triangular array of binomial coeﬃcients is called Pascal’s triangle after the seventeenth-century French mathematician Blaise Pascal.

Use the binomial theorem to prove that if A is a finite set, then |P (A)| = 2|A| 16. a A state’s lottery involves choosing six diﬀerent numbers out of a possible 36. How many ways can a person choose six numbers? b What is the probability of a person winning with one bet? 17. Use the binomial theorem to calculate 99983 . Hint. 9998 = 10000 2 18. In the card game Blackjack, there are one or more players and a dealer. Initially, each player is dealt two cards and the dealer is dealt one card down and one facing up.

0 , 1] has 2 elements . [2] has 1 elements . [0 , 2] has 2 elements . [1 , 2] has 2 elements . [0 , 1, 2] has 3 elements . [3] has 1 elements . [0 , 3] has 2 elements . [1 , 3] has 2 elements . [0 , 1, 3] has 3 elements . [2 , 3] has 2 elements . [0 , 2, 3] has 3 elements . [1 , 2, 3] has 3 elements . [0 , 1, 2, 3] has 4 elements . 3 A Exercises 1. Let A = {0, 2, 3}, B = {2, 3}, C = {1, 4}, and let the universal set be U = {0, 1, 2, 3, 4}. List the elements of a A⇥B e A ⇥ Ac c A⇥B⇥C g B3 b B⇥A d U ⇥; f B2 h B ⇥ P(B) 14 CHAPTER 1.