Five coins whose faces are marked 2, 3 are thrown. What is the probability of obtaining a total of 12?

Options

A. `5//32`
B. `11//16`
C. `5//16`
D. `10//16`

Answer

To solve the problem of finding the probability of obtaining a total of 12 when five coins marked with faces 2 and 3 are thrown, we can follow these steps:

### Step 1: Determine the total number of outcomes
When each coin can land on either 2 or 3, the total number of outcomes when throwing 5 coins is given by:
\[ \text{Total outcomes} = 2^5 = 32 \]

### Step 2: Identify the favorable outcomes for a total of 12
To obtain a total of 12 from five coins, we need to find combinations of 2s and 3s that sum to 12.

Let \( x \) be the number of coins showing 2, and \( y \) be the number of coins showing 3. We have the following equations:
1. \( x + y = 5 \) (since there are 5 coins)
2. \( 2x + 3y = 12 \) (since the total must be 12)

From the first equation, we can express \( y \) in terms of \( x \):
\[ y = 5 - x \]

Substituting this into the second equation:
\[ 2x + 3(5 - x) = 12 \]
\[ 2x + 15 - 3x = 12 \]
\[ -x + 15 = 12 \]
\[ -x = -3 \implies x = 3 \]

Substituting \( x = 3 \) back into the equation for \( y \):
\[ y = 5 - 3 = 2 \]

Thus, we need 3 coins showing 2 and 2 coins showing 3.

### Step 3: Calculate the number of favorable arrangements
The number of ways to arrange 3 coins showing 2 and 2 coins showing 3 can be calculated using the formula for combinations:
\[ \text{Number of arrangements} = \frac{5!}{3! \cdot 2!} \]
Calculating this:
\[ = \frac{120}{6 \cdot 2} = \frac{120}{12} = 10 \]

### Step 4: Calculate the probability
Now, we can find the probability of obtaining a total of 12:
\[ \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{10}{32} = \frac{5}{16} \]

### Final Answer
Thus, the probability of obtaining a total of 12 when throwing five coins is:
\[ \frac{5}{16} \]

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