If 4 dice ae rolled once, the numberof ways of getting the sum as 10 is K, then the value of `(K)/(10)` is equal to

Answer

To solve the problem of finding the number of ways to get a sum of 10 when rolling 4 dice, we can use a systematic approach.

### Step-by-Step Solution:

1. **Understanding the Problem**: We need to find the number of combinations of rolling 4 dice such that the sum of the numbers on the top faces equals 10. Each die can show a number from 1 to 6.

2. **Using Generating Functions**: The generating function for a single die is given by:
\[ x + x^2 + x^3 + x^4 + x^5 + x^6 = x(1 + x + x^2 + x^3 + x^4 + x^5) = x \frac{1 - x^6}{1 - x} \]
Therefore, for 4 dice, the generating function becomes:
\[ (x + x^2 + x^3 + x^4 + x^5 + x^6)^4 = \left(x \frac{1 - x^6}{1 - x}\right)^4 = x^4 (1 - x^6)^4 (1 - x)^{-4} \]

3. **Finding Coefficient of \(x^{10}\)**: We need to find the coefficient of \(x^{10}\) in the expansion of the generating function. This can be simplified to finding the coefficient of \(x^6\) in \((1 - x^6)^4 (1 - x)^{-4}\).

4. **Expanding the Functions**:
- The expansion of \((1 - x^6)^4\) can be done using the binomial theorem:
\[ (1 - x^6)^4 = \sum_{k=0}^{4} \binom{4}{k} (-1)^k x^{6k} \]
- The expansion of \((1 - x)^{-4}\) is given by:
\[ (1 - x)^{-4} = \sum_{n=0}^{\infty} \binom{n + 3}{3} x^n \]

5. **Combining the Expansions**: We need to find the coefficient of \(x^6\) in the product of these two series:
\[ \sum_{k=0}^{4} \binom{4}{k} (-1)^k \binom{6 - 6k + 3}{3} \]
This means we need to evaluate for \(k = 0, 1, 2, 3, 4\).

6. **Calculating Each Term**:
- For \(k = 0\): \(\binom{4}{0} \binom{9}{3} = 1 \cdot 84 = 84\)
- For \(k = 1\): \(-\binom{4}{1} \binom{3}{3} = -4 \cdot 1 = -4\)
- For \(k = 2\): \(\binom{4}{2} \binom{-3}{3} = 6 \cdot 0 = 0\) (since \(\binom{-3}{3} = 0\))
- For \(k = 3\): \(-\binom{4}{3} \binom{-9}{3} = 0\) (since \(\binom{-9}{3} = 0\))
- For \(k = 4\): \(\binom{4}{4} \binom{-15}{3} = 0\) (since \(\binom{-15}{3} = 0\))

7. **Summing Up**:
\[ K = 84 - 4 + 0 + 0 + 0 = 80 \]

8. **Finding \(\frac{K}{10}\)**:
\[ \frac{K}{10} = \frac{80}{10} = 8 \]

### Final Answer:
The value of \(\frac{K}{10}\) is \(8\).