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By Rudenskaya O. G.

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B) How many such words contain the letter a ? (c) How many contain the letters a and b ? (d) How many contain the letters b and c ? (e) How many contain the letters a, b and c ? (f) How many begin with a and end with b ? (g) How many begin with b and end with c ? Solution. (a) C(5, 3) C(21, 5) 8 ! ) (b) C(4, 2) C(21, 5) 8 ! (c) C(4, 2) C(20, 4) 8 ! (d) C(5, 3) C(19, 3) 8 ! (e) C(4, 2) C(19, 3) 8 ! (f) C(4, 2) C(20, 4) 6 ! (g) C(5, 3) C(19, 3) 6 !. 104. In how many ways can a committee of 10 be chosen so that there exactly 5 females and 3 juniors on the committee ?

When a word is formed from these letters, a letter may appear at the most the number of time it appear in the word committee or not at all. So generating function for c, o and i is given by (1 + x) each, whereas, for c, m and e is given by ⎛ x2 ⎞ ⎜⎜ 1 + x + ⎟ each. 2 ! ⎠ ⎝ 3 ⎛ x2 ⎞ 1 + x + ⎜⎜ ⎟ 2 ! ⎟⎠ ⎝ 3 If words are to be formed taking all the letters at once, then the numbers of such words is given by the coefficient of x9 9! and this is equal to . 9! 2! 64. A fair six-sided die is tossed four times and the numbers shown are recorded in a sequence.

A) We must choose three men from 20 and then two women from 12. 20 ⎛12 ⎞ The answer is ⎛⎜ ⎞⎟ ⎜ ⎟ = 1140(66) = 75,240. ⎝ 3 ⎠ ⎝ 2⎠ COUNTING PRINCIPLES AND GENERATING FUNCTIONS 29 (b) We calculate the case of four women and five women separately and add the results (using the addition rule). ⎛12 ⎞ ⎛ 20 ⎞ ⎛12 ⎞ ⎛ 20 ⎞ The answer is ⎜ ⎟ ⎜ ⎟ + ⎜ ⎟ ⎜ ⎟ = 495(20) + 792 = 10,692. 78. In how many ways can 20 students out of a class of 32 be chosen to attend class on a late Thursday afternoon (and take notes for the others) if (a) Paul refuses to go to class ?

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