Discrrete mathematics for computer science counting subset warmup
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Counting Subsets Warmup
We need a committee
• … with 3 professors and 2 students.
There are 15 professors and 100
students. How many different
committees could we form?
⎛ 15 ⎞ ⎛ ⎞
⎜⎝ 3 ⎟⎠ ⎜⎝ ⎟⎠
But wait!
• I forgot to say. Every committee has
to have a woman on it. There are 2
women professors and 35 women
students.
• OK, remove the committees with no
women.
⎛ 13 ⎞ ⎛ ⎞
⎜
⎜
⎟
⎟
⎝ 3 ⎠ ⎝ ⎠
But wait!
• I misspoke. There has to be a woman
professor AND a woman student.
There are 2 women professors and
35 women students.
⎛ 2 ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞
⎜
⎜
⎜
⎜
⎟
⎟
⎟
⎟
⎝ 1 ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
But wait!
• What I really need is to set up 5 committees A, B, C,
D, E with disjoint membership. How many ways?
⎛ 15 ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎝ 3 ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
• If rearranging the same committees to have
different names is not considered creating a
different set of committees, this number must be
divided by 25!
FINIS
