Explanation
1. Calculate Total Outcomes (n(S)):
When rolling four six-sided dice, each die has 6 possible outcomes. Since the dice are rolled independently, the total number of outcomes is:
n(S)=6×6×6×6=64=1296
2. Calculate Favorable Outcomes (n(E)):
We need the sum of the four dice to be equal to 6. We are looking for the number of integer solutions to:
x1+x2+x3+x4=6
where 1≤xi≤6 for each die.
To solve this, we can list the distinct combinations of digits that sum to 6:
Combination (1, 1, 1, 3): These digits can be arranged in 3!1!4!=4 ways.
Combination (1, 1, 2, 2): These digits can be arranged in 2!2!4!=6 ways.
Adding these permutations together gives the total number of favorable outcomes:
n(E)=4+6=10
3. Final Probability Calculation:
The probability P is the ratio of favorable outcomes to the total number of outcomes:
P=n(S)n(E)=129610
Simplifying the fraction by dividing the numerator and denominator by 2:
P=6485
Correct Option:
(c) 6485
