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 follow these steps:

### Step 1: Understand the Problem
When rolling 4 dice, each die can show a number from 1 to 6. We need to find the combinations of these numbers that add up to 10.

### Step 2: Adjust for Minimum Values
Since each die must show at least a 1, we can start by assigning 1 to each die. This means we initially have:
\[ d_1 + d_2 + d_3 + d_4 = 10 \]
If we assign 1 to each die, we can rewrite the equation as:
\[ (d_1 - 1) + (d_2 - 1) + (d_3 - 1) + (d_4 - 1) = 10 - 4 \]
This simplifies to:
\[ x_1 + x_2 + x_3 + x_4 = 6 \]
where \( x_i = d_i - 1 \) (i.e., \( x_i \) can be 0 to 5).

### Step 3: Use Stars and Bars Theorem
We need to find the number of non-negative integer solutions to the equation \( x_1 + x_2 + x_3 + x_4 = 6 \). The number of solutions can be found using the stars and bars theorem, which states that the number of ways to distribute \( n \) indistinguishable objects (stars) into \( r \) distinguishable boxes (bars) is given by:
\[ \binom{n + r - 1}{r - 1} \]

In our case, \( n = 6 \) and \( r = 4 \):
\[ \text{Number of solutions} = \binom{6 + 4 - 1}{4 - 1} = \binom{9}{3} \]

### Step 4: Calculate \( \binom{9}{3} \)
Calculating \( \binom{9}{3} \):
\[ \binom{9}{3} = \frac{9!}{3!(9-3)!} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = \frac{504}{6} = 84 \]

### Step 5: Exclude Invalid Cases
Now we need to exclude cases where any \( x_i > 5 \) (since \( x_i \) can only be 0 to 5). If, for example, \( x_1 = 6 \), then:
\[ x_1 - 6 + x_2 + x_3 + x_4 = 0 \]
This means \( x_2 = x_3 = x_4 = 0 \), which is one invalid case. Since there are 4 dice, there are 4 such cases to exclude (one for each die).

Thus, we subtract these 4 cases from our total:
\[ 84 - 4 = 80 \]

### Step 6: Find \( K \) and Calculate \( \frac{K}{10} \)
We found that \( K = 80 \). Now we need to calculate \( \frac{K}{10} \):
\[ \frac{K}{10} = \frac{80}{10} = 8 \]

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

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