- There are $3$ boys and $3$ girls. How many ways can they be arranged in a way that two girls won't be together?
Instead of making things harder, I directly write an assumption as illustrated below
$$\text{All permutations = Total possibilities – Two girls are together}$$
We already notice that total possibilities should be $6!$. Let's calculate the permutation of two girls are together. Recall that $K$ represents Girls, $E$ represents boys.
Entity $1$: $K_1K_2 \implies 1$
Entity $2$: $B_1B_2B_3K_3 \implies 4$
Entity $1$ and $2$: $5!$
And girls can be rearranged in $2$, which yields $5!\times 2!$. Plugging into the equation we wrote
$$6!-5!\times 2! = 480$$
However, the final answer I got should be wrong. Can you assist?
Best Answer
Method 1: We arrange the three boys in a row, then place the girls in the spaces between the boys and at the ends of the row to separate them.
The three boys can be arranged in a row in $3!$ ways. This creates four spaces, two between successive boys and two at the ends of the row. $$\square B_1 \square B_2 \square B_3 \square$$ To ensure that no two girls are adjacent, we must choose of these spaces in which to place the girls, which can be done in $\binom{4}{3}$ ways. The three girls can be arranged in the selected spaces in $3!$ ways. Hence, the number of admissible arrangements is $$\binom{4}{3}3!3!$$
Method 2: We correct your attempt by using the Inclusion-Exclusion Principle.
There are $6!$ ways to arrange the six people. As you stated, we wanted to exclude those arrangements in which a pair of girls are adjacent.
There are $\binom{3}{2}$ ways to select two girls who are adjacent. We have five objects to arrange, the block of two girls and the other four children. The objects can be arranged in $5!$ ways and the girls can be arranged within the block in $2!$ ways. Hence, there are $$\binom{3}{2}5!2!$$ arrangements in which two girls are adjacent.
If we subtract the number of arrangements in a pair of girls are adjacent from the total, we will have subtracted too much since we will have subtracted each arrangement in which there are two pairs of adjacent girls twice, once for each pair of adjacent girls we could have designated as the pair of adjacent girls. We only want to subtract those arrangements once, so we have to add the number of arrangements in which there are two pairs of adjacent girls to the total.
To have two pairs of adjacent girls, all three girls must be consecutive. If we treat the three consecutive girls as a block, we have four objects to arrange, the block of three girls and the three boys. The objects can be arranged in $4!$ ways. The girls can be arranged within the block in $3!$ ways. Hence, the number of arrangements in which there are two pairs of adjacent girls is $4!3!$.
Hence, the number of admissible arrangements is $$6! - \binom{3}{2}5!2! + \binom{3}{3}4!3!$$