<p>A group of 8 people consists of 4 married couples.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">The group stands in line. Find the number of different possible orders in which no two men stand next to each other.</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">The group stands in a circle. Find the number of different possible orders in which each man stands next to his wife.</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">The group forms a committee consisting of two teams of four people each. Find the number of ways that the committee can be formed such that neither team consists of only men or women.</span></div><p style="text-align: right;">[2]</p></li></ol>

A group of 8 people consists of 4 married couples.

The group stands in line. Find the number of different possible orders in which no two men stand next to each other.

The group stands in a circle. Find the number of different possible orders in which each man stands next to his wife.

The group forms a committee consisting of two teams of four people each. Find the number of ways that the committee can be formed such that neither team consists of only men or women.

Default solution

4 men and 4 women in a line, no two men adjacent.

Arrange 4 women: \(4!\)

Choose 4 of 5 gaps for the 4 men: \({}^{5}C_{4}\)

Arrange men in chosen gaps: \(4!\)

\(=4!\times{}^{5}C_{4}\times 4!=2880\)

Circle, each man next to his wife.

Treat each couple as a block. Arrange 4 blocks in a circle: \((4-1)!=6\)

Internal arrangement of each couple: \((2!)^{4}=16\)

\(=(4-1)!\times(2!)^{4}=96\)

Two teams of 4, neither team all men or all women.

Teams are identical (unordered).

No. of ways \(=\dfrac{{}^{8}C_{4}\times{}^{4}C_{4}}{2!}-{}^{4}C_{4}\times{}^{4}C_{4}\)

\(=35-1=34\)

2019 NJC P2 Q6

2019 NJC P2 Q6

Video Solution:

Default solution