<s>[INST] The number of distinct powers of x that appear in the expansion of $(x^{7}+x^{11}+x^{14})^{10}$ is 
The number of distinct powers of x that appear in the expansion of
$(x^{7}+x^{11}+x^{14})^{10}$ is

Let the power of $x^7$ is a, $x^{11}$ is b and $x^{14}$ is c
then by multinomial a+b+c=10
so we have to find the number of distinct values of $7*a+11*b+14*c= 7*(a+2c)+11*b$
how do I proceed after this?
 [/INST] First of all, notice that the answer to the question is equal to that of $(1+x^4+x^7)^{10}$, and then the answer is equal to the number of distinct sums of the form $4a+7b$, where $0\le a+b\le10$. If $a+b=k$, it is easy to notice that $4a+7b$ produces $k+1$ numbers, hence, we already have $1+2+...+11=66$ numbers. However some of these numbers are repeated while being produced. In other words;
$$4a+7b=4c+7d\implies 4(a-c)=7(d-b)\implies a-c=7.$$
Now, we have;
$$a=7, c=0\implies (d,b)=(4,0),(5,1),(6,2),or\; (7,3)$$
$$a=8, c=1\implies (d,b)=(4,0),(5,1), or\;(6,2)$$
$$a=9, c=2\implies (d,b)=(4,0), or\;(5,1)$$
$$a=10, c=3\implies (d,b)=(4,0).$$
Therefore, the answer is $66-10=56.$
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