To find the number that satisfies the given conditions, we can use the Chinese Remainder Theorem. Let's solve the problem step by step:

Step 1: Understanding the problem
We are looking for a number that leaves a remainder of 12 when divided by 13, a remainder of 11 when divided by 12, and a remainder of 10 when divided by 11.

Step 2: Expressing the conditions as congruences
We can express the given conditions as congruences (modular equations) as follows:
- x ≡ 12 (mod 13)
- x ≡ 11 (mod 12)
- x ≡ 10 (mod 11)

Step 3: Applying the Chinese Remainder Theorem
According to the Chinese Remainder Theorem, if the moduli (13, 12, and 11 in this case) are pairwise coprime (meaning they have no common factors), there exists a unique solution for the congruences.

Since 13, 12, and 11 are pairwise coprime, we can find the solution using the Chinese Remainder Theorem.

Step 4: Solving the congruences
To solve the congruences, we can use the method of successive substitution or the extended Euclidean algorithm. In this case, we will use the method of successive substitution:

Starting with the first congruence: x ≡ 12 (mod 13)
- The first number that satisfies this congruence is 12.
- The next numbers that satisfy this congruence repeat every 13 units. Therefore, the possible values for x are 12, 25, 38, 51, etc.

Now, let's substitute these values into the second congruence: x ≡ 11 (mod 12)
- We check which of the possible values for x from the previous step satisfies this congruence.
- By substituting the values, we find that 25 satisfies this congruence.

Finally, let's substitute this value into the third congruence: x ≡ 10 (mod 11)
- By substituting x = 25 into the third congruence, we find that it satisfies the congruence.

Therefore, the number that satisfies all the given conditions is x = 25.

Step 5: Matching the answer with the options
Out of the given options (1716, 1715, 1717, 716), only option B (1715) matches the number we found (x = 25).

Hence, the correct answer is option B (1715).