**Question:-**

Given a set {1, 2, 3, 5, 8, 13, 21, 34, 55}, how many integers between and including 3 and 89 cannot be represented as the sum of exactly two elements of this set?

**OPTIONS**

1) 34

2) 36

3) 51

4) 53

5) None of these

**Solution**The elements of the given set are distinct, and their least sum is 3, while maximum sum is 89.

Two numbers of this set, which contains 9 numbers, can be selected in 9C2 = 36 ways

We can see that each element of this set greater than 2 is a sum of two elements of the set. This means that the sum of any two elements of the set will be distinct from the sum of any other two elements of the set.

Thus, 36 different numbers between and including 3 and 89 can be formed by adding 2 numbers of the set.

Hence, the number of integers which cannot be represented as the sum of exactly two elements of this set will be 89 – 3 + 1 – 36 = 51.

Hence, option 3.

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