Note : This is not homework, it is self-study.An employer interviews eight people for four openings in the company. Three of the eight people are women. If all eight are qualified, in how many ways could the employer fill the four positions if a)the selection is random and b)exactly two are women?Part a) is already taken care of, now for part b). My reasoning goes like this:There must be 2 women chosen, so imagine that for the first position we choose a woman. There's 3 ways to choose. For the second position we choose another woman, there's 2 ways to choose. For the third position we choose a man, there's 5 ways to choose. For the fourth position we choose another man, there's 4 ways to choose. Then we multiply: 3 ⋅ 2 ⋅ 5 ⋅ 4 = 120. But the answer in the book is 30.

Question

Note : This is not homework, it is self-study.An employer interviews eight people for four openings in the company. Three of the eight people are women. If all eight are qualified, in how many ways could the employer fill the four positions if a)the selection is random and b)exactly two are women?Part a) is already taken care of, now for part b). My reasoning goes like this:There must be 2 women chosen, so imagine that for the first position we choose a woman. There&#039;s 3 ways to choose. For the second position we choose another woman, there&#039;s 2 ways to choose. For the third position we choose a man, there&#039;s 5 ways to choose. For the fourth position we choose another man, there&#039;s 4 ways to choose. Then we multiply:   3  ⋅  2  ⋅  5  ⋅  4  =  120. But the answer in the book is 30.

Rubi Boyle · Accepted Answer

You can&#039;t simply multiply like that to get the correct number of combinations. What you are doing gives you the number of permutations, but as in this case order does not matter, that is incorrect.Consider selecting 2 women from a group of 3 women. Your logic suggests there are six different possible combinations, which is clearly incorrect as there are 3 combinations (as order does not matter). There are however, six permutations.Once you&#039;ve established the difference between the two and notice you require a combination for this question, your result is given by                              (                            3        2                              )                      ⋅                              (                            5        2                              )                      =  30

herbariak1 · Accepted Answer

Your calculation assumes that the positions are distinguishable: that putting persons   A,  B,  C, and   D in positions 1,2,3, and 4, respectively, is different from putting person   A in position 2, person   B in position 1, and persons   C and   D in positions 3 and 4, respectively. That is also how I would read the question.

Note : This is not homework, it is self-study. An employer interviews eight people for four ope

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