If three dices are thrown then the probability that the sum of the numbers on their uppoermost faces to be atleast 5 is

Options

A. `(1)/(53)`
B. `(53)/(54)`
C. `(1)/(54)`
D. `(52)/(53)`

Explanation

To solve the problem of finding the probability that the sum of the numbers on the uppermost faces of three dice is at least 5, we can follow these steps:

### Step-by-Step Solution:

1. **Determine the Total Outcomes:**
When three dice are thrown, each die has 6 faces. Therefore, the total number of outcomes when throwing three dice is:
\[ \text{Total outcomes} = 6^3 = 216 \]

**Hint:** Remember that each die operates independently, so multiply the number of outcomes for each die.

2. **Find the Outcomes for the Sum Less Than 5:**
We need to find the number of outcomes where the sum of the numbers on the uppermost faces of the three dice is less than 5. The possible sums that are less than 5 are 3 and 4.

- **Sum = 3:** The only combination is (1, 1, 1). This is 1 way.
- **Sum = 4:** The combinations are:
- (1, 1, 2)
- (1, 2, 1)
- (2, 1, 1)

Each of these combinations can occur in 3 different arrangements (permutations). So, there are 3 ways.

Therefore, the total number of outcomes where the sum is less than 5 is:
\[ \text{Total outcomes for sum < 5} = 1 + 3 = 4 \]

**Hint:** Consider how many different ways you can arrange the numbers for each sum.

3. **Calculate the Probability of Sum Less Than 5:**
The probability of the sum being less than 5 is given by the ratio of favorable outcomes to total outcomes:
\[ P(\text{sum} < 5) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{4}{216} = \frac{1}{54} \]

**Hint:** Always simplify your fractions to their lowest terms.

4. **Calculate the Probability of Sum At Least 5:**
The probability of the sum being at least 5 is the complement of the probability of the sum being less than 5. Therefore:
\[ P(\text{sum} \geq 5) = 1 - P(\text{sum} < 5) = 1 - \frac{1}{54} = \frac{54 - 1}{54} = \frac{53}{54} \]

**Hint:** The complement rule is a powerful tool in probability; it can simplify your calculations significantly.

### Final Answer:
The probability that the sum of the numbers on the uppermost faces of the three dice is at least 5 is:
\[ \frac{53}{54} \]