Animated Solution for Mathematics - Binomial Theorem: For natural numbers $m,n$ if $(1-y{)}^m(1+y{)}^n=1+a_{1}y+a_2{y}^2+....$ and $a_1=a_2=10$, then $(m,n)$ is

Welcome, fellow traveler, to the fascinating world of binomial expansions! We are given the expression $(1-y{)}^m(1+y{)}^n=1+a_{1}y+a_2{y}^2+\dots$ and told that $a_1=a_2=10$.

Our mission is to find the natural numbers $m$ and $n$. Let's embark on this journey together.

The first trap many students fall into is trying to expand the entire product. We only care about the coefficients of $y$ and $y^2$.

Recall the standard binomial expansion:

Applying this to our two binomials, we get:

Now, let's multiply these two expansions. We only need terms that result in $y$ and $y^2$.

For $a_1$ (the coefficient of $y$), we multiply the constant $1$ from the first bracket with $ny$ from the second, and $-my$ from the first with the constant $1$ from the second. This gives us $a_1=n-m$.

Since $a_1=10$, we have our first equation:

Now for the main event: $a_2$ (the coefficient of $y^2$). This comes from three pairings: the constant $1$ with the $y^2$ term of the second bracket, the $y$ term of the first with the $y$ term of the second, and the $y^2$ term of the first with the constant $1$ of the second.

Summing these, we get:

This expression for $a_2$ simplifies by taking a common denominator of $2$:

Rearranging the terms, we identify the perfect square $(n-m{)}^2$ and the remaining linear terms:

Since $a_2=10$ and $n-m=10$, we substitute these values:

This simplifies to $100-(n+m)=20$, which means $n+m=80$. Now we have a simple system of equations:

Adding these equations gives $2n=90$, so $n=45$. Subtracting them gives $2m=70$, so $m=35$.

We have arrived at our destination: $(m,n)=(35,45)$.