Counting Subsets Warmup We need a committee • … with professors and students There are 15 professors and 100 students How many different committees could we form? ⎛ 15 ⎞ ⎛ ⎞ ⎜⎝ ⎟⎠ ⎜⎝ ⎟⎠ But wait! • I forgot to say Every committee has to have a woman on it There are women professors and 35 women students • OK, remove the committees with no women ⎛ 13 ⎞ ⎛ ⎞ ⎜ ⎜ ⎟ ⎟ ⎝ ⎠ ⎝ ⎠ But wait! • I misspoke There has to be a woman professor AND a woman student There are women professors and 35 women students ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ But wait! • What I really need is to set up committees A, B, C, D, E with disjoint membership How many ways? ⎛ 15 ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ • 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 ... professors and 100 students How many different committees could we form? ⎛ 15 ⎞ ⎛ ⎞ ⎜⎝ ⎟⎠ ⎜⎝ ⎟⎠ But wait! • I forgot to say Every committee has to have a woman on it There are