Ytukyg

Find the number of ways of arranging the letters $text{AAAAA, BBB, CCC, D, EE & F}$ in a row if no two $text{C's}$ are together?

My Attempt:

Well, I should I arrive at the answer if I subtract the cases where 3 $text{C's}$ and 2 $text{C's}$ appear together from the total.

Total possibilities $= frac{15!}{5!3!3!2!}$

Total possibilities where 3 $text{C's}$ appear $=frac{13!}{5!3!2!}$

However, I am not able to find the possibilities for 2 $text{C's}$ being together and get to the answer.

Any help would be appreciated.

While I prefer the approach explained by Alaleh A and Parcly Taxel, here is how you can solve the problem using the Inclusion-Exclusion Principle.

The number of distinguishable arrangements of $5$ As, $3$ Bs, $3$ Cs, $1$ D, $2$ Es, and $1$ F is
$$frac{15!}{5!3!3!1!2!1!}$$
as you found.

From these, we must subtract those arrangements in which a pair of Cs are adjacent.

A pair of Cs are adjacent: We have $14$ objects to arrange, $5$ As, $3$ Bs, $1$ CC, $1$ C, $1$ D, $2$ Es, and $1$ F. They can be arranged in
$$frac{14!}{5!3!1!1!1!2!1!}$$
ways.

However, if we subtract $frac{14!}{5!3!1!1!1!2!1!}$ from the total, we will have subtracted too much since we will have subtracted each arrangement with two pairs of adjacent Cs twice (such arrangements have three consecutive Cs), once when we designated the first two Cs as the adjacent pair and once when we designated the last two Cs as the adjacent pair. Since we only want to subtract such arrangements once, we must add them back.

Two pairs of adjacent Cs: As mentioned above, this means the three Cs are consecutive. Hence, we have $13$ objects to arrange, $5$ As, $3$ Bs, $1$ CCC, $1$ D, $2$ Es, and $1$ F. The number of such arrangements is
$$frac{13!}{5!3!1!1!2!1!}$$
as you found.

Hence, the number of arrangements of $5$ As, $3$ Bs, $3$ Cs, $1$ D, $2$ Es, and $1$ F in which no two Cs are consecutive is
$$frac{15!}{5!3!3!1!2!1!} - frac{14!}{5!3!1!1!2!1!} + frac{13!}{5!3!1!1!2!1!}$$

Ignore the C's for the moment and arrange the remaining letters. There are $frac{12!}{5!3!2!}=332640$ ways to do this.

Now consider the thirteen spaces between and beyond the letters – at most one C may be inserted into each space, so given an arrangement of the other letters there are $binom{13}3=286$ ways to add C's.

The total number of admissible combinations is thus $332640×286=95135040$.

Try a simpler approach, start with counting he number of ways you can arrange all letters except C's. it's:

$$ frac{12!}{5!3!2!}$$

now there are 13 places to put the C's where they won't be next to each other so the answer is:

$$frac{12!}{5!3!2!} binom{13}{3}$$

Let $n_1$ be all letters except C ($=12$)and $n_2$ be Cs ($=3$). I am going to generalize the solution in such a manner that you have no two C's together (p=0), two C's together (p=1) and three C's together (p=2)

Each pair must have a left-hand member, which we can choose from any of the Cs except the last: $tbinom{n_2-1}{p}$. This will create $n_2-p$ blocks of Cs, which we can distribute into the $n_1+1$ slots between and around the non-Cs: $tbinom{n_1+1}{n_2-p}$. Thus the number of arrangemenets with exactly $p$ pairs is $tbinom{n_2-1}{p}tbinom{n_1+1}{n_2-p}.frac{n_1!}{5!.3!.2!}$

For p = 0 in the question

$$tbinom{3-1}{0}tbinom{12+1}{3-0}.frac{12!}{5!.3!.2!}$$

$$tbinom{2}{0}tbinom{13}{3}.frac{12!}{5!.3!.2!}$$

First arrange all the letters without $C$'s, i.e. $$frac{12!}{5!3!1!2!1!}.$$
Now we have to fill inside the gaps between arranged letters.
As they given no two $C$'s come together, means $3$ $C$'s also doesn't come together.
The answer:
$$
frac{12!}{5!3!1!2!1!}*frac{P^{13}_3}{3!}.
$$
For example I have $3$ chocolates, that doesn't mean that I don't have $2$.