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Chapter 1. Counting

Chapter 1. Counting - 1 1.1 Counting Basic principle total...

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§ 1 Counting § 1.1 Basic principle 1.1.1 A simple rule: total k jobs n 1 ways to do job 1 n 2 ways to do job 2 . . . n k ways to do job k ( n 1 n 2 · · · n k ) ways to do the k jobs § 1.2 Selection of distinguishable objects 1.2.1 Definition. For any integer n 0, define the factorial of n , written n !, by n ! = ( 1 , n = 0 , n ( n - 1) · · · 1 , n 1 . 1.2.2 Theorem. (Binomial Series Theorem) For any real number α and for any | x | < 1, (1 + x ) α = 1 + α x + 1 2! α ( α - 1) x 2 + 1 3! α ( α - 1)( α - 2) x 3 + · · · . In particular, if α = n is a positive integer, we have (1 + x ) n = 1 + n 1 x + n 2 x 2 + · · · + n n x n , where n r = n ! r !( n - r )! is known as a binomial coefficient . 1.2.3 Ordered selection without replacement Problem : n distinguishable objects, choose r from them without replacement in an ordered sequence. Example : Beauty Contest, 15 candidates, how many ways to award Winner, 1st and 2nd runners-up? Answer: 15 |{z} W inner × 14 |{z} 1 st × 13 |{z} 2 nd 3
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General case: n ( n - 1) · · · ( n - r + 1) = n ! ( n - r )! . Equivalent problem : allocate r distinguishable objects into n distinguishable cells, with no cell containing > 1 objects. 1.2.4 Unordered selection without replacement Problem : n distinguishable objects, choose r from them without replacement in an unordered sequence. Example : Beauty Contest, 15 candidates, how many ways to select 5 to enter final round? Answer: 15 5 General case: Consider a product of n terms (1 + x )(1 + x ) · · · (1 + x ) = (1 + x ) n = 1 + n 1 x + n 2 x 2 + · · · + n n x n . No. of ways to select r from n objects in an unordered sequence = no. of ways to select r x ’s from the n brackets above = coefficient of x r in expansion of (1 + x ) n .
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