In how many different ways can 22 be written as the sum of 3 different prime numbers? That is, determine the number of triples $(a, b, c)$ of prime numbers with $1<a<b<c$ and $a+b+c=22$.
Every prime number other than 2 is odd.

Since $a, b$ and $c$ are all prime and $a+b+c=22$ which is even, it cannot be the case that all of $a, b$ and $c$ are odd (otherwise $a+b+c$ would be odd).

Thus, at least one of $a, b$ and $c$ is even.

Since $1<a<b<c$, then it must be the case that $a=2$ and $b$ and $c$ are odd primes.

Since $a=2$, then $b+c=20$.

Since $b$ and $c$ are primes with $b<c$, then $b=3$ and $c=17$, or $b=7$ and $c=13$.

Therefore, there are two triples $(a, b, c)$ of primes numbers that satisfy the requirements.

ANSWER: 2