1. A poker hand is a 5-card hand dealt randomly from a deck. (a) How many poker
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Question
1. A poker hand is a 5-card hand dealt randomly from a deck.
(a) How many poker hands contain exactly three hearts?
(b) How many poker hands contain at least two kings?
2. Our Xena fan club has 14 male and 17 female members. We must select a president, vice-president and treasurer.
(a) How many ways are there to assign these jobs?
(b) In how many ways can we assign the jobs, such that a man becomes treasurer?
3. A six-letter word shall mean any string of six letters; for instance, BZQYQR counts. (We do not insist that it be a word in any particular language.)
(a) How many six-letter words are there?
(b) How many six-letter words contain at least one K?
Explanation / Answer
(1)
(A)
3 hearts can be selected in 13C3 ways and remaining 2 cards can be selected in 39C2 ways.
Hence total possible poker hands with exactly 3 hearts are 13C3*39C2 = 211926
(B)
2, 3 or 4 kings can be selected in 4C2 or 4C3 or 4C4 ways respectively.
Number of poker hands with at least 2 kings = (4C2 * 48C3) + (4C3 * 48C2) + (4C4 * 48C1) = 108336
(2)
(A)
There are total 14+17 = 31 members in the fan club. One can select any 3 members for the required positions.
This can be done in 31C3 = 4495 ways
(B)
If the treasurer has to be man, this can be done in 14C1 = 14 and remaining 2 members can be selected in 30C2 ways.
Hence total possibilities are 14C1 * 30C2 = 6090
(3)
(A)
Total number of possible 6 letter words are = 26^6
(B)
Number of 6 letter words with at least one K = 26^6 - (25^6)
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