How many summands are there I have some problem understanding this Exercise/problem.
What is summand ? I have searched for it, but nothing concrete came up.
Problem:
Look at the multinomial theorem. How many summands are
there in $(x+y+z)^7$ and in $(w+x+2y+z)^9$ ?
Can someone explain to me what it is(summand) and how to get the solution.
The solution should be 36 for the first one and 220 for the last one.
 A: The number of summands is determined by $\binom{n+k-1}{k-1}$ or $\binom{n+k-1}{n}$ where $n$ is the exponent and $k$ is the number of unknowns ($x,y,z...$)
A summand for $(x+y+z)^7$ might be $210x^3y^2z^2$ or $x^7y^0z^0$ (or just written $x^7$)
A: A summand is an expression in a sum. 
In $a+b$, $a$ and $b$ are the summands.
A: If you were to multiply out the expressions and combine like terms, you'd arrive at the number of summands.
The number of summands in the first is the number of triples $(a,b,c)$, with $0 \leq a,b,c \in \mathbb{Z} \leq 7$ and $a+b+c=7$.  (Do you see why?)
The number of summands in the second is the number of quadruples $(a,b,c,d)$, with $0 \leq a,b,c,d \in \mathbb{Z} \leq 9$ and $a+b+c+d=9$.  (Do you see why?)
EDIT: As an example, let's do $(a+b+c)^3$.
Multiplied out (I used Wolfram Alpha), it's:
$$a^3 + b^3 + c^3 + 3a^2b + 3a^2c + 3b^2c + 3b^2a + 3c^2a + 3c^2b + 6abc.$$
Now, let's look at the exponents of $a,b,c$ on each summand, in order:
$$(3,0,0), (0,3,0), (0,0,3), (2,1,0), (2,0,1), (0,2,1), (1,2,0), (1,0,2), (0,1,2), (1,1,1).$$
I listed all of the triples $(a,b,c)$ for which $0 \leq a,b,c \in \mathbb{Z} \leq 3$ and $a+b+c=3$.
Hopefully this makes it a bit clearer.
