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Candelabra

Problem

Three white candles and two black candles can be arranged in a number of ways in a pentagon shaped candelabra.


If the candles are placed at random, find the probability that the three white candles will be adjacent.


Solution

This can be solved in two different ways.

Method 1
Begin by numbering each of the candle holders:


We can use these labels to indicate which candle goes in which place:

B B W W W *
B W B W W
B W W B W
B W W W B *
W B B W W *
W B W B W
W B W W B
W W B B W *
W W B W B
W W W B B *

Due to the circular arrangement we need to be careful in identifying the arrangements for which whites are adjacent (marked with *).

As there are ten possible ways of filling the five candle holders, the probability of three whites being adjacent is 5/10 = 1/2.

Method 2
Without loss of generality we can consider a black candle being placed in the top circle; if this is not the case, it is always possible to rotate the arrangement so that this is true.


It can be seen that if the other black candle is placed in 1 or 4, three whites will be adjacent.

Therefore, the probability of three whites being adjacent is 2/4 = 1/2.

If four white candles and two black candles were placed in a hexagonal candelabra, what would be the probability of the whites being adjacent?
Investigate for different numbers of black and white candles.

Problem ID: 183 (01 Nov 2004)     Difficulty: 2 Star

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