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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 Fall '08
 Burbidge,John
 Marble

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