The number of ways in which the letters AAABBCD can be arranged such that all A's are together is?

Steps to Solve

1. Treat the A's as a single unit

Since all three 'A's must be together, we can treat them as a single block or unit. So instead of thinking of 'AAABBCD', think of it as 'XBBCD', where 'X' represents 'AAA'.

2. Count the total units

Now we have the units: X, B, B, C, D. That gives us a total of 5 units: X, B, B, C, D.

3. Calculate arrangements considering repeated letters

Since the letter 'B' appears twice, the formula for the arrangements is given by the factorial of the total units divided by the factorial of the repeated units.

The formula is:

$$ \text{Arrangements} = \frac{n!}{p_1! \cdot p_2! \cdots p_k!} $$

In this case, it becomes:

$$ \text{Arrangements} = \frac{5!}{2!} $$

4. Calculate factorials

Calculating the factorials,

• $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$
• $2! = 2 \times 1 = 2$

5. Final calculation of arrangements

Now calculate the number of arrangements:

$$ \text{Arrangements} = \frac{120}{2} = 60 $$

So there are 60 different ways to arrange the letters of 'AAABBCD' with all A's together.