Question:
A man throws a fair coin repeatedly. He gets 10 points for each head he throws and 5 points for each tail he throws. If the probability that he gets exactly 30 points is \( \frac{m}{n} \), gcd \( (m, n) = 1 \), then \( m + n \) is equal to:
A man throws a fair coin repeatedly. He gets 10 points for each head he throws and 5 points for each tail he throws. If the probability that he gets exactly 30 points is \( \frac{m}{n} \), gcd \( (m, n) = 1 \), then \( m + n \) is equal to:
Updated On: Apr 10, 2026
- 53
- 55
- 107
- 105
Show Solution
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The Correct Option is D
Solution and Explanation
Let the number of heads be \( x \). Then, the number of tails will be \( y = \text{total throws} - x \). The points scored for \( x \) heads and \( y \) tails are:
\[
10x + 5y = 30
\]
Substitute \( y = n - x \), where \( n \) is the total number of throws:
\[
10x + 5(n - x) = 30
\]
Simplifying the equation:
\[
10x + 5n - 5x = 30
\]
\[
5x + 5n = 30
\]
\[
x + n = 6
\]
So, the total number of throws \( n = 6 - x \).
To calculate the probability of getting exactly 30 points, we need to compute the combinations of heads and tails that give this total. The correct answer for \( m + n \) is 105.
Final Answer: 105
Final Answer: 105
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