Let's break down each part of the question based on the image:

Given:

Seven letters are drawn from the set: $D, M, S, T, A, E, O$.
The vowels in this set are $A, E, O$, and the consonants are $D, M, S, T$.

Part (a):

Find the number of arrangements of the letters.

Since all the letters are distinct, the number of arrangements (or permutations) of these 7 letters is given by: $7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$

Part (b):

Find the number of arrangements with the three vowels together.

If the three vowels $A, E, O$ must be together, treat them as a single block. Now, you are left with 5 items to arrange: the 4 consonants and the "vowel block." These 5 items can be arranged in: $5! = 5 \times 4 \times 3 \times 2 \times 1$ Additionally, within the "vowel block," the three vowels can be arranged among themselves in: $3! = 3 \times 2 \times 1$ Thus, the total number of arrangements is: $5! \times 3!$

Part (c):

Find the number of arrangements with the three vowels all separated.

To find the number of arrangements where the vowels are separated, you can first arrange the consonants and then place the vowels in the gaps between them. There are 4 consonants, and when arranged, they create 5 possible positions (gaps) for the vowels: one before the first consonant, one between each pair of consonants, and one after the last consonant.

You need to choose 3 of these 5 gaps to place the vowels, which can be done in: $\binom{5}{3} = \frac{5!}{3!(5-3)!} = 10 \text{ ways.}$ Once the gaps are chosen, the vowels can be arranged in: $3! = 6 \text{ ways.}$ Thus, the total number of arrangements is: $4! \times \binom{5}{3} \times 3!$

Let's calculate all of these.

Part (a): $7!$

$7! = 5040$

Part (b): $5! \times 3!$

$5! = 120, \quad 3! = 6, \quad 5! \times 3! = 120 \times 6 = 720$

Part (c): $4! \times \binom{5}{3} \times 3!$

$4! = 24, \quad \binom{5}{3} = 10, \quad 3! = 6, \quad 24 \times 10 \times 6 = 1440$

Final answers:

• (a) $5040$
• (b) $720$
• (c) $1440$

Would you like further details or clarifications on any part?

Here are 5 related questions to expand your understanding:

1. How would the answers change if there were repeated letters?
2. What is the formula for combinations and when is it used in counting problems?
3. How can the principle of inclusion-exclusion be applied to counting arrangements?
4. What if only 5 of the 7 letters were used, how would that affect the arrangements?
5. How do the positions of the vowels change if only two vowels must be together?

Tip: When solving permutation problems, always verify if the elements are distinct or if there are constraints (such as some letters must be together). This can drastically change the counting method.