Alist contains twenty integers, not necessarily distinct. does the list contain at least two consecutive integers?

(1) if any single value in the list is increased by 1, the number of different values in the list does not change.
(2) at least one value occurs more than once in the list.

a. statement (1) alone is sufficient, but statement (2) alone is not sufficient.
b. statement (2) alone is sufficient, but statement (1) alone is not sufficient.
c. both statements together are sufficient, but neither one alone is sufficient.
d. each statement alone is sufficient. statements (1) and (2) together are not sufficient.

Answer c is correct. Both statements arent sufficient separately, but together they are sufficient.

Step-by-step explanation:

Neither of the statments are sufficient alone. Lets first analyze why statment (1) alone is not sufficient.

If we suppose that our list is scattered, so our values are pretty distant one to the other, one value increasing by one wont change this. The list will still be scattered and our values will remain distant one to the other.

For a concrete example, consider a list with values multiples of a thousand. Our list can look this way {1000,2000,3000, , 19000, 20000}. So we have 20 distinct values. Adding one to any of this 20 values wont change the fact that we have 20 different values. However, our list doesnt have 2 consecutive integers.

Proving (2) not being sufficient is pretty straightfoward. If our list contains 20 equal values it wont have 2 consecutive values, because all values are equal. For example the list with values {0, 0, 0, 0, , 0} will have the value 0 occuring more than once but it doesnt have 2 consecutive values.

Now, lets assume both statements (1) and (2) are valid.

Since (2) is valid there is an integer, lets call it k, such that k appears on the list at least twice. Because (1) is True, then if we take one number with value k and we increase its value by one, then the number of distinct values shoudnt change. We can observe a few things:

By combining points 2 and 3, we deduce that the lists doesnt lose values, because point 2 tells us that the only possible value to be lost is k, and point 3 says that the value k will be preserved!

Since the list doesnt lose values and the number of different values is the same, we can conclude that the list shoudnt gain values either, because the only possibility for the list to adquire a new value after adding one to a number is to lost a previous value because the number of distinct numbers does not vary!

Point 4 tells us that value k+1 was obtained on the new list after adding 1 to k. We reach the conclusion that the new list doesnt have new values from the original one, that means that k+1 was alredy on the original list.

Thus, the original list contains both the values k (at least twice) and k+1 (at least once), therefore, the list contains at least two consectutive values.

answer: $201

step-by-step explanation: to get the answer, you first get 3 nieces that he sold and multiply it by 67 and get $201, so he earned $201 selling his nieces.