Question

A circular birthday cake is being adorned with candles, equally placed around in a circle on the top of the cake. There are fourteen candles in total to be placed, each having a different colour. In how many ways can the candles be placed if the black candle must be between the beige candle and the yellow candle?

Answer 1

We treat the black, beige, and yellow candles as one unit and the other candles as separate units to use the formula for circular permutations. The total arrangements of the candles is 11! * 2.

Step-by-step explanation:

The subject of this question is combinatorial mathematics, specifically, the arrangement of objects in a circle also referred as circular permutations. To solve this problem, we first consider the black, beige, and yellow candles as a single unit since the black candle must be between the beige and yellow candles. Therefore, we have 12 'units' (the three candles as one unit, and the other 11 candles). The number of ways these 'units' can be arranged in a circle is (n-1)!, so it is (12-1)! = 11!.

However, within the unit of the black, beige, and yellow candles, they can be arranged in 3! ways. Because the black candle has to be between the beige and the yellow candles, there are 2 ways of arranging these three.

Therefore, the total number of ways the candles can be arranged is 11! * 2.

Learn more about Circular Permutations here: