There are $N$ sequences with 15 terms and the following properties:

- each term is an integer,
- at least one term is between -16 and 16 , inclusive,
- the 15 terms have at most two different values,
- the sum of every six consecutive terms is positive, and
- the sum of every eleven consecutive terms is negative.

The value of $N$ is
(A) 48
(B) 72
(C) 64
(D) 80
(E) 56
Since the terms in each such sequence can be grouped to get both positive and negative sums, then there must be terms that are positive and there must be terms that are negative.

Since the 15 terms have at most two different values and there are terms that are positive and terms that are negative, then the terms have exactly two different values, one positive and one negative.

We will call these values $x$ and $y$. We know that both are integers and we will add the condition that $x>0$ and $y0$. Since the sum of eleven consecutive terms is always negative, then $a_{1}+a_{2}+a_{3}+a_{4}+a_{5}+a_{6}+$ $a_{7}+a_{8}+a_{9}+a_{10}+a_{11}0$ and $\left(a_{1}+a_{2}+a_{3}+a_{4}+a_{5}+a_{6}\right)+\left(a_{7}+a_{8}+a_{9}+a_{10}+a_{11}\right)0$.

Since $a_{7}+a_{8}+a_{9}+a_{10}+a_{11}0$, then $a_{6}>0$ and so $a_{6}=x$.

The six term condition also tells us that $a_{7}+a_{8}+a_{9}+a_{10}+a_{11}+a_{12}>0$.

Since $a_{7}+a_{8}+a_{9}+a_{10}+a_{11}0$ which gives $a_{12}=x$.

We can repeat this argument by shifting all of the terms one further along in the sequence.

Starting with $a_{2}+a_{3}+a_{4}+a_{5}+a_{6}+a_{7}>0$ and $a_{2}+a_{3}+a_{4}+a_{5}+a_{6}+a_{7}+a_{8}+a_{9}+a_{10}+a_{11}+a_{12}0$ and $a_{5}+a_{6}+a_{7}+a_{8}+a_{9}+a_{10}+a_{11}+a_{12}+$ $a_{13}+a_{14}+a_{15}0)$.

Also, the sum of the first 11 terms in the sequence is $10 x+y$ which is negative (that is, $10 x+y0$ and so $10 x+y=5 x+(5 x+y)>0$, which contradicts $10 x+y0$.

Also, the sum of each group of 11 consecutive terms is $9 x+2 y$.

We are told that $9 x+2 y0$ and $y0$ and $9 x+2 y0$ and $9 x+2 y0$, then $x>-0.2 y$. Since $y \geq-16$, then $x\frac{2}{9}(1)=\frac{2}{9}$.

Since $\frac{2}{9}<x<3.2$, then any sequence with $-16 \leq y \leq-1$ also has $1 \leq x \leq 16$, and so we have counted this already.

This means that $N$, the total number of such sequences, equals 56 .

ANswer: (E)