Question

In how many ways can $3$ ladies and $4$ gentleman arrange themselves around a round table so that no two ladies are together?

• A. $60$
• B. $80$
• C. $20$
• D. $144$

Answer 1

Answer: (D)

$144$

No women sit together.

As number of women is less than number of men , two of the men will be in between a pair of women.

So fix the seats of women. Women can be arranged in $3!=6$ ways . Now select two men. They can be selected in ${}^4{C}_2=6$ ways and arranged in $2!=2$ ways. Consider these $2$ men as a single man. Now we have $3$ men and they can be arranged in the round table in $(3-1)!=2!=2$ ways .

Therefore total number of ways$=6\times 6\times 2\times 2=144$ ways .