106
COMBINATIONS
[CHAP. 17
(b) One group of four can be selected in 12 C4 ways, then another in 8 C4 ways, and the third in 1 way. Since the order in which the groups are formed is now immaterial, the division may be made in 12 C4 ·8C4 · 1 4 3! ¼ 5775 ways.
17.8
The (a) (b) (c) (d )
English alphabet consists of 21 consonants and 5 vowels. In how many ways can 4 consonants and 2 vowels be selected ? How many words consisting of 4 consonants and 2 vowels can be formed ? How many of the words in (b) begin with R ? How many of the words in (c) contain E ? (a)
The 4 consonants can be selected in 21 C4 ways and the 2 vowels can be selected in 5 C2 ways. Thus, the selections may be made in 21 C4 ·5C2 ¼ 59 850 ways.
(b)
From each of the selections in (a), 6! words may be formed by permuting the letters. Therefore, 59850 · 6! ¼ 43 092 000 words can be formed.
(c)
Since the position of the consonant R is fixed, we must select 3 other consonants (in 20C3 ways) and 2 vowels (in 5 C2 ways), and arrange each selection of 5 letters in all possible ways. Thus, there are 20 C3 ·5C2 · 5! ¼ 1 368 000 words.
(d ) Since the position of the consonant R is fixed but the position of the vowel E is not, we must select 3 other consonants (in 20 C3 ways) and 1 other vowel (in four ways), and arrange each set of 5 letters in all possible ways. Thus, there are 20 C3 · 4 · 5! ¼ 547 200 words.
17.9
From an ordinary deck of playing cards, in how many different ways can five cards be dealt (a) consisting of spades only, (b) consisting of black cards only, (c) containing the four aces, (d ) consisting of three cards of one suit and two of another, (e) consisting of three kings and a pair, ( f ) consisting of three of one kind and two of another? (a) From the 13 spades, 5 can be selected in
13 C5
¼ 1287 ways.
(b) From the 26 black cards, 5 can be selected in 26C5 ¼ 65 780 ways. (c) One card must be selected from the 48 remaining cards. This can be done in 48 different ways. (d ) A suit can be selected in four ways and three cards from the suit can be selected in 13 C3 ways; a second suit can now be selected in three ways and two cards of this suit in 13 C2 ways. Thus, three cards of one suit and two of another can be selected in 4 ·13C3 · 3 ·13C2 ¼ 267 696 ways. (e) Three kings can be selected from the four kings in 4 C3 ways, another kind can be selected in 12 ways, and two cards of this kind can be selected in 4 C2 ways. Thus, three kings and another pair can be dealt in 4 C3 · 12 ·4C2 ¼ 288 ways. ( f ) A kind can be selected in 13 ways and three of this kind in 4 C3 ways; another kind can be selected in 12 ways and two of this kind can be selected in 4 C2 ways. Thus, 3 of one kind and 2 of another can be dealt in 13 ·4C3 · 12 ·4C2 ¼ 3744 ways.
17.10
(a) Prove: The total number of combinations of n objects taken successively 1; 2; 3; . . . ; n at a time is 2n 1. (b) In how many different ways can one invite one or more of five friends to the movies. (a) The total number of combinations is n C1
þ nC2 þ nC3 þ · · · þ nCn ¼ 2n 1
since, from Problem 42.8(a), n C0
þ nC1 þ nC2 þ · · · þ nCn ¼ 2n
(b) The number is 5 C1 þ 5C2 þ 5C3 þ 5C4 þ 5C5 ¼ 25 1 ¼ 31: