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# A_Problem - 1 If 5 black 7 red 9 blue and 6 white marbles...

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1 If 5 black, 7 red, 9 blue, and 6 white marbles are arranged at random, what is the probability that every white marble is adjacent to at least one other white marble? Solution- Let us represent the colors black, red, blue and white by BLA, R, B and W. We consider disjoint events that guarantee the desired event asked for in the Problem. Case 1- Let the 6 white marble glued together. Call it as W 0 . So, we have 5 BLA, 7 R, 9 B and 1 W 0 . These can be arranged in N 1 = 22! 5!7!9!1! different ways. Case 2- Let there be 4 W together apart from 2 other W that are also together. Let us call the 4 W that are together by W 0 and the 2 W that are also together by W 00 . Then, we have 5 BLA, 7 R, 9 B, 1 W 0 and 1 W 00 . These can be arranged in 23! 5!7!9!1!1! . But, this also considers the cases where W 0 W 00 or W 00 W 0 happens. Remember that we want W 0 and W 00 be apart. By the previous case, the number of times that each of the cases W 0 W 00 or W 00 W 0 happens is N 1 . Therefore, the number of ways that case 2 happens is N 2 = 23!

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A_Problem - 1 If 5 black 7 red 9 blue and 6 white marbles...

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