There are 455 ways in which the product of the three prime numbers will result in an even number.
How to find the number of ways
To find the number of ways in which the product of three randomly chosen prime numbers from the first 15 primes will result in an even number, consider the factors that contribute to an even product.
An even number is divisible by 2. So, to have an even product, include at least one factor of 2 in the prime numbers chosen.
Among the first 15 prime numbers, only one prime number is 2. Therefore, in order to have an even product, choose at least one 2 among the three primes.
Now let's consider the possibilities:
Selecting 2 as one of the primes:
We have one choice for the prime number 2.
We need to choose 2 more primes from the remaining 14 primes (excluding 2).
The number of ways to choose 2 primes from 14 is given by the binomial coefficient (14 choose 2), which can be calculated as C(14, 2) = 91.
Not selecting 2 as any of the primes:
We have 14 odd primes to choose from (excluding 2).
We need to choose 3 primes from these 14 primes.
The number of ways to choose 3 primes from 14 is given by the binomial coefficient (14 choose 3), which can be calculated as C(14, 3) = 364.
Therefore, the total number of ways in which the product of three randomly chosen prime numbers from the first 15 primes will result in an even number is 91 + 364 = 455.
Hence, there are 455 ways in which the product of the three prime numbers will result in an even number.