MST10040_2013_Lec7(1) - Example Two ways of approaching the...

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Example Two ways of approaching the same question. Question Out of 5 men and 7 women, a committee of 2 men and 3 women is to be formed. In how many ways can this be done if one man and one woman are married and cannot both be on the committee? Answer: First method. If the given man and woman are both on the committee, we can choose the remaining people in 4 1 × 6 2 = 60 ways. If we subtract this number from the total number of ( 5 2 ) × ( 7 3 ) = 350 possibilities, we get the number of possibilities with at most one of the member from the pair. Answer is 350 - 60 = 290. Lecture 7 MST10040 October 3, 2013 1 / 9
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Example Answer: Second method. First suppose that the given man and woman are both not on the committee, thus we choose 2 men from 4 men and 3 women from 6 women in ( 4 2 ) × ( 6 3 ) = 120 ways. Second, we can also have that the given man is on the committee and thus the given woman should not be on the committee. Hence we choose 1 man from 4 men and 3 women from 6 women in ( 4 1 ) × ( 6 3 ) = 80 ways. Third, we can also have that the given woman is on the committee and thus the given man should not be on the committee. Hence we choose 2 men from 4 men and 2 women from 6 women in ( 4 2 ) × ( 6 2 ) = 90 ways. Summing up all the possibilities, we obtain 120+80+90=290 as before. Lecture 7 MST10040 October 3, 2013 1 / 9
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Counting techniques Consider the word S U C C E S S F U L L Y . Question How many different arrangements are there of the 12 letters of this word? Answer: If the letters were different, we could rearrange them in 12 ! ways, forming 12 ! “words”. Lecture 7 MST10040 October 3, 2013 2 / 9
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Counting techniques Consider the word S U C C E S S F U L L Y . Question How many different arrangements are there of the 12 letters of this word? Answer: Let’s look at some of those, concentrating on the letters S: C E S 1 S 2 F U Y L L S 3 U C C E S 2 S 1 F U Y L L S 3 U C C E S 3 S 2 F U Y L L S 1 U C . . . Lecture 7 MST10040 October 3, 2013 2 / 9
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Counting techniques Consider the word S U C C E S S F U L L Y . Question How many different arrangements are there of the 12 letters of this word? Answer: Treating the 3 occurrences of the letter S as different gives rise to too many copies of the word C E S S F U Y L L S U C .
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