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, In a standard set of double-six dominoes, there is exactly one domino for every possible (unordered) pair of numbers from 0 to 6 (including doubles, i.e. a domino with two fives.) A hand of dominoes is another word for a set of dominoes. (a) How many dominoes are there (show your work)? (b) How many five-domino hands are there? (c) How many five-domino hands contain no domino with a 0? (d) How many five-domino hands contain at least one double? (e) How many five-domino hands contain no domino with a value greater than 3? (f) How many dominoes are in a standard set of double-n dominoes (where there is exactly one domino for every possible unordered pair of numbers from 0 to n including doubles.)

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Answer #1

(a)

It is known that domino have two values separated by a line. The given set of 7 elements is 0,1,2,3,4,5,6; Select a pair of numbers from the above set to form a domino. Thus, the number of dominoes with different numbers 2 There will be 7 dominoes with same values. Therefore, the number of dominoes can be formed from the set of double-6 set+7-28 2 The 28 dominoes are as follows:

Hand referred as set of dominoes. There are total 28 dominoes. Select set of five dominoes from the set of 28 dominoes. Thus,

(c)

There are total 28 dominoes. There are 7 dominoes (0-0,0-1,0-2,0-3,0-4,0-5 and 0-6) out of 28 dominoes has 0. Thus, there are 28-7-21 dominoes without 0. Select five dominoes from the set of these 21 dominoes. Therefore, the number of set of five dominoes- 551.16! 21x 20x19 x18x17 x16! 5x4x3x2x16! 21×19×3×17 -20349

There are seven out of 28 dominoes are doubles (0-0,1-1,2-2,3-3,4-4,5-5 and 6-6). Thus, there are 28-7-21 dominoes without doubles. By part b, there are totaets of five-dominoes. 28 21 By part c, there are total sets of five-dominoes without a double. To get sets of five-dominoes with at least one double, subtract total number of sets of five-dominoes with no double from the total number of sets of five-dominoes . (28)-(?) Thus, the number of five-domino hands with at least one double 98280 20349 -77931

There are 10 dominos (0-0,0-1,0-2,0-3,1-1,1-2,1-3,2-2,2-3 and 3-3) out of 28 dominoes have values less than or equal to 3 Need to select set of any five dominoes from these 10 dominoes number of sets of five-dominos with values less than 3- (1010 551.5! 10x9x8x7x6x5! 5×4×3×2×51 -9x2x 7x 2 - 252

(ђ A pair of numbers can be selected from set { 0,1,2, ,n} of n+1 numbers = Also, there will be n+1 number of doubles(0-0,1-1,2-2,..., n-n) The number of dominoes in a standard set of double-n dominoes- 2 ntnt1 (n+1)×nx(n-1) +(n+1 2x(n-1) n+1)xn +(n+1

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