The last digit of the following expreesion is :
`(1!)^1 + (2!)^(2) + (3!)^(3) + (4!)^(4) + …(10!)^(10)`

Options

A. 4
B. 5
C. 6
D. 7

Answer

To find the last digit of the expression \( (1!)^1 + (2!)^2 + (3!)^3 + (4!)^4 + \ldots + (10!)^{10} \), we will calculate each term individually and then sum them up, focusing only on the last digit of each term.

### Step-by-step Solution:

1. **Calculate \( (1!)^1 \)**:
\[ 1! = 1 \implies (1!)^1 = 1^1 = 1 \]
Last digit: **1**

2. **Calculate \( (2!)^2 \)**:
\[ 2! = 2 \implies (2!)^2 = 2^2 = 4 \]
Last digit: **4**

3. **Calculate \( (3!)^3 \)**:
\[ 3! = 6 \implies (3!)^3 = 6^3 = 216 \]
Last digit: **6**

4. **Calculate \( (4!)^4 \)**:
\[ 4! = 24 \implies (4!)^4 = 24^4 \]
To find the last digit of \( 24^4 \), we only need the last digit of 24, which is 4.
\[ 4^4 = 256 \]
Last digit: **6**

5. **Calculate \( (5!)^5 \)**:
\[ 5! = 120 \implies (5!)^5 = 120^5 \]
The last digit of 120 is 0, so:
\[ 120^5 \text{ has a last digit of } 0. \]
Last digit: **0**

6. **Calculate \( (6!)^6 \)**:
\[ 6! = 720 \implies (6!)^6 = 720^6 \]
The last digit of 720 is 0, so:
\[ 720^6 \text{ has a last digit of } 0. \]
Last digit: **0**

7. **Calculate \( (7!)^7 \)**:
\[ 7! = 5040 \implies (7!)^7 = 5040^7 \]
The last digit of 5040 is 0, so:
\[ 5040^7 \text{ has a last digit of } 0. \]
Last digit: **0**

8. **Calculate \( (8!)^8 \)**:
\[ 8! = 40320 \implies (8!)^8 = 40320^8 \]
The last digit of 40320 is 0, so:
\[ 40320^8 \text{ has a last digit of } 0. \]
Last digit: **0**

9. **Calculate \( (9!)^9 \)**:
\[ 9! = 362880 \implies (9!)^9 = 362880^9 \]
The last digit of 362880 is 0, so:
\[ 362880^9 \text{ has a last digit of } 0. \]
Last digit: **0**

10. **Calculate \( (10!)^{10} \)**:
\[ 10! = 3628800 \implies (10!)^{10} = 3628800^{10} \]
The last digit of 3628800 is 0, so:
\[ 3628800^{10} \text{ has a last digit of } 0. \]
Last digit: **0**

### Final Calculation:
Now we sum the last digits we calculated:
\[ 1 + 4 + 6 + 6 + 0 + 0 + 0 + 0 + 0 + 0 = 17 \]
The last digit of 17 is **7**.

### Conclusion:
The last digit of the expression \( (1!)^1 + (2!)^2 + (3!)^3 + (4!)^4 + \ldots + (10!)^{10} \) is **7**.