[Math] Seven people (3 women and 4 men) arrange them selves randomly in seven consecutive seats in a row…

probability

Seven people (3 women and 4 men) arrange them selves randomly in seven consecutive seats in a row, find the probability the women will be in three adjacent seats.

How to do this problem?

Best Answer

There are $7!$ total ways of arranging the people.

If all three women sit together, then there are $3!$ ways of arranging just the women.

Then, the block of women and the four men need to get arranged. There are $5!$ ways to do this, as the block of women can be treated as one object to arrange, giving a total of five.

Thus, there are $3! * 5!$ ways that the people can be arranged such that the three women are together.

Therefore, the probability that the three women are together is $\frac{3! * 5!}{7!} = \frac{6 * 5!}{7 * 6 * 5!} = \frac{1}{7}$.