Question

Seven men and seven women line up at a checkout counter in a store. In how many ways can they line up if the first person in line is a man , and the people in line alternate man, woman, man, woman ,and so on

Answer 1

Answer:  The required number of ways is 25401600.

Step-by-step explanation:  Given that seven men and seven women line up at a checkout counter in a store.

We are to find the number of ways in which they can line up, if the first person in line is a man , and the people in line alternate man, woman, man, woman and so on.

Since the first person is a man, so

for first position, we have 7 options.

There are alternate man and women in the line, so the second person must be a women.

That is, we have 7 options for the second position.

Similarly, for 3rd and 4th positions, we have 6 options for each, and so on.

Therefore, the number of ways in which the 14 persons can line up is

$n=7\times7\times6\times6\times5\times5\times4\times4\times3\times3\times2\times2\times1\times1=7!\times7!=5040\times5040=25401600.$

Thus, the required number of ways is 25401600.