Answer:
a) 6
Step-by-step explanation:
Expanding the polynomial using the formula:
[tex]$(x+y)^n=\sum_{k=0}^n \binom{n}{k} x^{n-k} y^k $[/tex]
Also
[tex]$\binom{n}{k}=\frac{n!}{(n-k)!k!}$[/tex]
I think you mean [tex]210x^6y^4[/tex]
We can deduce that this term will be located somewhere in the middle. So I will calculate [tex]k= 5; k=6 \text{ and } k =7[/tex].
For [tex]k=5[/tex]
[tex]$\binom{10}{5} (y)^{10-5} (x)^{5}=\frac{10!}{(10-5)! 5!}(y)^{5} (x)^{5}= \frac{10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5! }{5! \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 } \\ =\frac{30240}{120} =252 x^{5} y^{5}$[/tex]
Note that we actually don't need to do all this process. There's no necessity to calculate the binomial, just [tex]x^{n-k} y^k[/tex]
For [tex]k=6[/tex]
[tex]$\binom{10}{6} \left(y\right)^{10-6} \left(x\right)^{6}=\frac{10!}{(10-6)! 6!}\left(y\right)^{4} \left(x\right)^{6}=210 x^{6} y^{4}$[/tex]
