哈工大深圳组合数第六章习题答案.pdf

HW3 1. Find the number of integers between 1 and 10,000 inclusive that are not divisible by 4, 6, 7, or 10. Solution: Let P1 be the property that an integer is divisible by 4, P2 be the property that an integer is divisible by 6, so we also get P , P 3 4 which are related to 7 and 10.Let S be the set consisting of the first 10000 positive integers. For i = 1,2,3,4 let A be the set consisting of i those integers in S with property P . We wish to find the number of i integers in A A A A . 1 2 3 4 A1 = 2500 A2 = 1666 A3 = 1428 A4 = 1000 A A = 833 A A = 357 A A = 500 1 2 1 3 1 4 A3 A2 = 238 A4 A2 = 333 A3 A4 =142 A A A = 119 A A A = 166 1 2 3 1 2 4 A A A = 47 A A A = 71 2 3 4 1 4 3 A A A A = 23 1 2 3 4 S = 10000 A A A A = S - Ai + A A - 1 2 3 4 i i A A A + A A A A = 5429 i j k 1 2 3 4 2. Determine the number of integral solutions of the equation x x x x 20 1 2 3 4 that satisfy 4 x 8 1x1  6 , 0 x2  7 , 3 , 2 x4  6 Solution: