Counting Ill Theorem 1(Binomial Theorem). For all n e nand a,be R: (a+bn= The expression(k)is often called a"binomial coefficient"in honor of its appearance here This reasoning about binomials extends nicely to multinomials, which are sums of two rmore terms. For example, suppose we wanted the coefficient of b02k2epr in the expansion of(6+o+k+e+p+r)l. Each term in this expansion is a product of 10 variables where each variable is one of b, o, k, e, p, or r. Now, the coefficient of bo-kepr is the number of those terms with exactly 1 b, 2o's, 2ks, 3es, 1 p, and 1 r. And the number of such terms is precisely the number of rearrangments of the word BOOKKEEpER 1122111!1(1,2,2,3,1,1 The expression on the left is an example of a"multinomial coefficient"and the notation on the right is a shorthand. This reasoning extends to a general theorem Theorem 2(Multinomial Theorem). For all n e nand 21,.Im ER: k+.+km=n You'll be far better off if your remember the reasoning behind the multinomial rem rather than this monstrous equation 3 Poker hands There are 52 cards in a deck. Each card has a suit and a value There are four suits And there are 13 values: 3,4,5,6,7,8,9,,J K. A Thus, for example, 89 is the 8 of hearts and Ad is the ace of spades. Values farther to the right in this list are considered"higher"and values to the left are"lower� � � � � � � 4 Counting III Theorem 1 (Binomial Theorem). For all n ∈ N and a, b ∈ R: �n � �n n−kbk (a + b) n = a k k=0 n The expression is often called a “binomial coefficient” in honor of its appearance here. k This reasoning about binomials extends nicely to multinomials, which are sums of two or more terms. For example, suppose we wanted the coefficient of 3 bo2 k2 e pr in the expansion of (b + o + k + e + p + r)10. Each term in this expansion is a product of 10 3 variables where each variable is one of b, o, k, e, p, or r. Now, the coefficient of bo2k2e pr is the number of those terms with exactly 1 b, 2 o’s, 2 k’s, 3 e’s, 1 p, and 1 r. And the number of such terms is precisely the number of rearrangments of the word BOOKKEEPER: 10! 10 = 1! 2! 2! 3! 1! 1! 1, 2, 2, 3, 1, 1 The expression on the left is an example of a “multinomial coefficient” and the notation on the right is a shorthand. This reasoning extends to a general theorem: Theorem 2 (Multinomial Theorem). For all n ∈ N and z1, . . . zm ∈ R: (z1 + z2 + . . . + zm) n = n z1 k1 z2 k2 . . . z km k1,...,km∈N k1, k2, . . . , km m k1+...+km =n You’ll be far better off if your remember the reasoning behind the Multinomial Theo￾rem rather than this monstrous equation. 3 Poker Hands There are 52 cards in a deck. Each card has a suit and a value. There are four suits: spades hearts clubs diamonds ♠ ♥ ♣ ♦ And there are 13 values: jack queen king ace 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , J , Q , K , A Thus, for example, 8♥ is the 8 of hearts and A♠ is the ace of spades. Values farther to the right in this list are considered “higher” and values to the left are “lower