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# Six boys and six girls are to sit in a row at random. The probability that boys and girls sit alternately is$A.{\text{ }}\dfrac{{6! \times {}^7{P_6}}}{{12!}} \\ B.{\text{ }}\dfrac{{6!6!}}{{12!}} \\ C.{\text{ }}\dfrac{{2 \times 6!6!}}{{12!}} \\ D.{\text{ }}\dfrac{{2 \times 6!}}{{12!}} \\$

Last updated date: 23rd Jul 2024
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Hint: First find out how many combinations of boys and girls sitting alternately are possible. Out of all the possible combinations girls would have 6! Ways because no. of girls are 6. But boys will have 7! (by using the concept of permutation) ways because the last boy can have two places either before the first girl or after the last girl and then accordingly calculate the probability.

$n\left( s \right) = 12!$ ways where S is sample space.
${G_1},{G_2},{G_3},{G_{_4}},{G_5},{G_6}$
6 boys can be seated in 7 places in ${}^7{P_6}$ ways.
$\therefore$ $n\left( E \right) = {\text{ 6! }} \times {}^7{{\text{P}}_6}$
$\therefore$ The required probability = $\dfrac{{n\left( E \right)}}{{n\left( S \right)}}$= $\dfrac{{6!\; \times {}^7{P_6}}}{{12!}}$
Note: The formula for calculating the permutation is given by $P(n,r) = \dfrac{{n!}}{{(n - r)!}}$, where n is the total no. of object in the set, r is the no. of choosing object from the set.