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How many numbers among 1, 2, 3, ..., 1000 are not divisible by 9?


How many distinct arrangements can be made from the letters in the word "REARRANGE''?


Two boys and three girls are going to sit around a table with 5 different chairs. If the two boys want to sit together, in how many possible ways can they be seated?


What fraction of all the 10-digit numbers with distinct digits have the property that the sum of every pair of neighboring digits is odd?


The product of all digits of positive integer M is 105 How many such Ms are there with distinct digits?

 Apr 27, 2019

\(\text{1000 is not divisible by 9}\\ \dfrac{999}{9}=111 \text{ numbers in 1-999 that are divisible by 9}\\ 1000-111 = 889 \text{ numbers that are not divisible by 9}\)


for the second question do you mean any length arrangement or 9 letter arrangements


\(\text{I assume you mean the chairs are distinct and thus the chair choice matters}\\ \text{There are 5 ways to choose 2 adjacent chairs}\\ \text{There are 2 ways to orient the 2 boys in those two adjacent chairs} \text{That gives us 10 ways to place the boys}\\ \text{then there are }3!=6 \text{ ways to arrange the girls in the 3 remaining chairs}\\ \text{That gives us }10\cdot 6 = 60 \text{ ways to arrange the 5 people as described}\)


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 Apr 27, 2019

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