Problem 4: Prove this strong version of the pigeonhole property. let 9, 92,...,qm be positive integers....

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Problem 4: Prove this "strong" version of the pigeonhole property. let 9₁, 92,...,qm be positive integers. If 91 +92 + +9m-m+1 objects are placed into m boxes, then the first box must contain q: objects, or the second box must contain 92 objects, ..., or the last box must contain qm objects. Problem 5: Suppose 50 socks lie are in a drawer. Each one is either white or black, anklehigh or knee- high, and either has a hole or doesn't. 22 socks are white, four of these have a hole, and one of these four is knee-high. Ten white socks are knee-high, ten black socks are knee-high, and five knee-high socks have a hole. Exactly three ankle-high socks have a hole. • Use inclusion and exclusion to determine the number of black, ankle-high socks with no holes. • Draw a Venn diagram that shows the number of socks with each combination of characteristics. Problem 6: How many integers between 1 and 1,000 are not divisible by of 5, 6 or 9? Problem 7: Use inclusion and exclusion to count the number of ways to rearrange "math is fun" where the words "math", "is", and "fun" are not in the arrangement. le we are not counting "ismathfun" or "innumathsf". Problem 4: Prove this "strong" version of the pigeonhole property. let 9₁, 92,...,qm be positive integers. If 91 +92 + +9m-m+1 objects are placed into m boxes, then the first box must contain q: objects, or the second box must contain 92 objects, ..., or the last box must contain qm objects. Problem 5: Suppose 50 socks lie are in a drawer. Each one is either white or black, anklehigh or knee- high, and either has a hole or doesn't. 22 socks are white, four of these have a hole, and one of these four is knee-high. Ten white socks are knee-high, ten black socks are knee-high, and five knee-high socks have a hole. Exactly three ankle-high socks have a hole. • Use inclusion and exclusion to determine the number of black, ankle-high socks with no holes. • Draw a Venn diagram that shows the number of socks with each combination of characteristics. Problem 6: How many integers between 1 and 1,000 are not divisible by of 5, 6 or 9? Problem 7: Use inclusion and exclusion to count the number of ways to rearrange "math is fun" where the words "math", "is", and "fun" are not in the arrangement. le we are not counting "ismathfun" or "innumathsf".

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Problem 4 To prove the strong version of the pigeonhole property we can use contradiction Suppose that none of the m boxes contains q1 q2 or qm objects Then the first box must contain at most q11 obje... View the full answer