In how many ways can a selection be made of 5 letters out of 5As, 4Bs, 3Cs, 2Ds and 1E?

Options

A. 70
B. 71
C. `""^(15)C_(5)`
D. None of these

Answer

To solve the problem of selecting 5 letters from the given quantities of letters (5 As, 4 Bs, 3 Cs, 2 Ds, and 1 E), we will analyze the different combinations of letters we can select based on their quantities.

### Step-by-Step Solution:

1. **Identify the total letters available:**
- A: 5
- B: 4
- C: 3
- D: 2
- E: 1

2. **Consider different cases based on the number of letters selected:**
- We will consider cases based on how many of each letter we can select while ensuring the total is 5.

3. **Case 1: All letters are the same (5 As)**
- Selection: AAAAA
- Count: 1 way

4. **Case 2: 4 of one letter and 1 of another**
- 4 As and 1 B: AAAA B → 1 way
- 4 As and 1 C: AAAA C → 1 way
- 4 As and 1 D: AAAA D → 1 way
- 4 As and 1 E: AAAA E → 1 way
- 4 Bs and 1 A: BBBB A → 1 way
- 4 Bs and 1 C: BBBB C → 1 way
- 4 Bs and 1 D: BBBB D → 1 way
- 4 Bs and 1 E: BBBB E → 1 way
- Total for this case: 8 ways

5. **Case 3: 3 of one letter and 2 of another**
- 3 As and 2 Bs: AAA BB → 1 way
- 3 As and 2 Cs: AAA CC → 1 way
- 3 As and 2 Ds: AAA DD → 1 way
- 3 Bs and 2 As: BBB AA → 1 way
- 3 Bs and 2 Cs: BBB CC → 1 way
- 3 Bs and 2 Ds: BBB DD → 1 way
- 3 Cs and 2 As: CCC AA → 1 way
- 3 Cs and 2 Bs: CCC BB → 1 way
- Total for this case: 8 ways

6. **Case 4: 3 of one letter, 1 of another, and 1 of a third**
- 3 As, 1 B, 1 C: AAA B C → 3!/(1!1!1!) = 6 ways
- 3 As, 1 B, 1 D: AAA B D → 6 ways
- 3 As, 1 B, 1 E: AAA B E → 6 ways
- 3 As, 1 C, 1 D: AAA C D → 6 ways
- 3 As, 1 C, 1 E: AAA C E → 6 ways
- 3 As, 1 D, 1 E: AAA D E → 6 ways
- 3 Bs, 1 A, 1 C: BBB A C → 6 ways
- 3 Bs, 1 A, 1 D: BBB A D → 6 ways
- 3 Bs, 1 A, 1 E: BBB A E → 6 ways
- 3 Bs, 1 C, 1 D: BBB C D → 6 ways
- 3 Bs, 1 C, 1 E: BBB C E → 6 ways
- 3 Bs, 1 D, 1 E: BBB D E → 6 ways
- Total for this case: 36 ways

7. **Case 5: 2 of one letter, 2 of another, and 1 of a third**
- 2 As, 2 Bs, 1 C: AA BB C → 5 ways (5!/2!2!1!)
- 2 As, 2 Bs, 1 D: AA BB D → 5 ways
- 2 As, 2 Bs, 1 E: AA BB E → 5 ways
- 2 As, 2 Cs, 1 B: AA CC B → 5 ways
- 2 As, 2 Cs, 1 D: AA CC D → 5 ways
- 2 As, 2 Cs, 1 E: AA CC E → 5 ways
- 2 Bs, 2 Cs, 1 A: BB CC A → 5 ways
- 2 Bs, 2 Cs, 1 D: BB CC D → 5 ways
- 2 Bs, 2 Cs, 1 E: BB CC E → 5 ways
- Total for this case: 30 ways

8. **Case 6: 2 of one letter, 1 of another, and 2 of a third**
- 2 As, 1 B, 1 C, 1 D: AA B C D → 4!/(2!1!1!) = 12 ways
- 2 As, 1 B, 1 C, 1 E: AA B C E → 12 ways
- 2 As, 1 B, 1 D, 1 E: AA B D E → 12 ways
- 2 As, 1 C, 1 D, 1 E: AA C D E → 12 ways
- 2 Bs, 1 A, 1 C, 1 D: BB A C D → 12 ways
- 2 Bs, 1 A, 1 C, 1 E: BB A C E → 12 ways
- 2 Bs, 1 A, 1 D, 1 E: BB A D E → 12 ways
- 2 Bs, 1 C, 1 D, 1 E: BB C D E → 12 ways
- Total for this case: 96 ways

9. **Case 7: All letters different (1 A, 1 B, 1 C, 1 D, 1 E)**
- Selection: A B C D E → 1 way

10. **Total the ways from all cases:**
- Case 1: 1
- Case 2: 8
- Case 3: 8
- Case 4: 36
- Case 5: 30
- Case 6: 96
- Case 7: 1
- Total = 1 + 8 + 8 + 36 + 30 + 96 + 1 = 181

### Final Answer:
The total number of ways to select 5 letters from the given quantities is **181**.