14 points

the number of integers between 500 and 700 such that the sum of their digits is 12 is

a) 11
b) 12
c) 13
d) 15
e) 20

Let's break down the question:

The number of integers between 500 and 700 such that the sum of their digits is 12

(I removed that last 'is' for the question for now.)

So, we are looking for how many integers are between 500 and 700, and when you find the sum of their digits, you get 12.

Let's find all those numbers, and see how many numbers we end up with!

507
516
525
534
543
552
561
570

606
615
624
633
642
651
660

So, now count how many numbers we have come up with.

We have 15 numbers!

So, now let's look at the whole question!

The number of integers between 500 and 700 such that the sum of their digits is 12 is:

The question is asking how many integers are there between 500 and 700 such that the sum of their digits is 12.

The question would have been better asked as, instead of what was given to you:

How many integers are between 500 and 700 such that the sum of their digits is 12?

So, we know that there are 15 integers between 500 and 700 such that the sum of the digits is 12.

The number of integers between 500 and 700 such that the sum of their digits is 12 is 15.

I hope you understand. If you have any more questions about your question, feel free to ask. :)