Algebraic Graph Theory by N. Biggs

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  • February 26, 2017
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By N. Biggs

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63. How many different words can be made from letters of the word ‘committee’. Solution. The word committee contains letters c, o and i once and m, t and e twice. 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!

Also define 0 ! = 1, note that 1 ! = 1 Thus, 4! 1, 6 ! 4 = 7! 7 . 6 . 5 . 4 . 3 . 1 = 3! 3 . 1 We read n ! as “n factorial”. It is true that 4 ! = 24 and 6 ! = 720 but frequently we leave our answers in factorial form rather than evaluating the factorials. Nevertheless, the relation n ! ] enables us to compute the values of n ! for small n fairly quickly. For example : 0 ! = 1, 1 ! =2 3 ! = 6, 4 ! = 24, 5 ! = 120 6 ! = 720, 7 ! = 5040, 8 ! = 40320 9 ! = 362880, 10 ! = 3628800, 11 ! = 39916800. 10.

Hiscock has ten children but his car holds only five people (including driver). When he goes to the circus, in how many ways can he select four children to accompany him ? Solution. The question involves choosing, not order. 10 ! 10 = 210 different ways. 6! 8. C(n, r) = C(n, n – r). Proof. (Algebraic) : We have that C(n, n – r) = n! ( n − r ) ! Simplifying, we find that C(n, n – r) = n! , which in turn is equal to C(n, r), as was to be shown. ( n − r ) ! 9. C(n, k) = C(n – 1, k) + C(n – 1, k – 1) for n > k > 0.

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