<s>[INST] Calculating value of a variable in combination? Using the formula (bionomial theorem):
$${}_nC_r \cdot a^{n-r} \cdot b^{r}$$
i was trying to find out the term with the coefficient $400$ from $(2x + 5y)^5$? Then i came up with this equation which i have no idea to solve beside trying numbers between 1 and 5. Is there any other way?
$$2^{5-r} \cdot 5^{r} \cdot {}_5C_r = 400$$
can we find the value of r without brute forcing?
 [/INST] $\frac{2^5}{2^r} \cdot 5^r \cdot \binom 5r = 400$
$\frac{32}{2^r} \cdot 5^r \cdot \binom 5r = 400$
$\frac{1}{2^r} \cdot 5^r \cdot \binom 5r = \frac{25}{2}$
$\frac{1}{2^r} \cdot 5^r \cdot \binom 5r = \frac{5 \cdot 5}{2}$
$\frac{1}{2^r} \cdot 5^r \cdot \binom 5r = \frac{5}{2} \cdot \binom 51$
On comparing both sides,
$2^r = 2^1$
$r = 1$
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