To determine which statements about the letter tiles are true, let's evaluate each one based on the combinations and probability involved:
We have 5 vowels (A, E, I, O, U) and 21 consonants (the rest of the letters).
A. There are 10 ways to choose 3 vowels.
- To choose 3 vowels from 5, we use the combination formula: \[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \] where \( n = 5 \) (vowels), and \( r = 3 \): \[ \binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5 \times 4}{2 \times 1} = 10 \] This statement is true.
B. There are 60 ways to choose 3 vowels.
- As calculated above, there are 10 ways to choose 3 vowels, not 60. This statement is false.
C. There are 2,600 ways to choose 3 letters.
- To find ways to choose 3 letters from the total of 26, we use: \[ \binom{26}{3} = \frac{26!}{3!(26-3)!} = \frac{26 \times 25 \times 24}{3 \times 2 \times 1} = \frac{15600}{6} = 2600 \] This statement is false.
D. There are 15,600 ways to choose 3 letters.
- Using the combination calculation: \[ \binom{26}{3} = 15600 \] This statement is true.
E. The probability that all 3 letters will be vowels is approximately 0.02308.
- The total ways to choose 3 letters is 15600 (as calculated). The ways to choose 3 vowels is 10. So, the probability is: \[ P(\text{3 vowels}) = \frac{\text{Ways to choose 3 vowels}}{\text{Total ways to choose 3 letters}} = \frac{10}{15600} = \frac{1}{1560} \approx 0.000641 \] This is far less than 0.02308. This statement is false.
The two correct statements are:
- A. There are 10 ways to choose 3 vowels.
- D. There are 15,600 ways to choose 3 letters.