$$ 26 \% $$ of college students say they use credit cards because of the rewards program. You randomly select 10 college students and ask each to name the reason he or she uses credit cards. Find the probability that the number of college students who say they use credit cards because of the rewards program is (a) exactly two, (b) more than two, and (c) between two and five inclusive. If convenient, use technology to find the probabilities. (a) $$ \mathrm{P}(2)=\square $$ (Round to the nearest thousandth as needed.)

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UpStudy AI · Accepted Answer

To find the probability that exactly two out of ten randomly selected college students use credit cards because of the rewards program, we can use the **binomial probability formula**:

$$
P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}
$$

where:
- $$ n = 10 $$ (the number of trials),
- $$ k = 2 $$ (the number of successes),
- $$ p = 0.26 $$ (the probability of success on a single trial).

### Step-by-Step Calculation:

1. **Calculate the binomial coefficient:**
   $$
   \binom{10}{2} = \frac{10!}{2!(10-2)!} = 45
   $$

2. **Calculate $$ p^k $$:**
   $$
   0.26^2 = 0.0676
   $$

3. **Calculate $$ (1-p)^{n-k} $$:**
   $$
   0.74^8 \approx 0.089925
   $$
   
   *(This was calculated step-by-step for accuracy.)*

4. **Combine all parts:**
   $$
   P(X = 2) = 45 \times 0.0676 \times 0.089925 \approx 0.274
   $$

### Final Answer:
$$
\boxed{0.274}
$$
The probability that exactly two out of ten college students use credit cards because of the rewards program is approximately 0.274.

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