451 /www/nrich/html/content/03/01/15plus2/ 1 2011-02-01T00:00:01 <mdoxml version="1.0"><br></br> Prove that the sum<br></br> <br></br> $$ \sum_{t=0}^m {(-1)^t\over t!(m-t)!} = 0 $$ <br></br></mdoxml> <mdoxml version="1.0"><div>This neat solution came from Marcos and the result was also proved by Yatir Halevi:</div> <div>By the binomial expansion:</div> <br></br> <div>$$(1+x)^m=\sum_{t=0}^m \frac{m!}{t!(m-t)!}x^t $$</div> <br></br> <div>This can be proved by induction on $m$ but I won't clutter this with unnecessary proofs.</div> <br></br> <div>Putting in $x= -1$ we have</div> <br></br> <div>$$0=\sum_{t=0}^m \frac{m!}{t!(m-t)!}(-1)^t $$</div> <br></br> <div>Dividing through by $m!$ gives us the required result:</div> <br></br> <div>$$\sum_{t=0}^m \frac{(-1)^t}{t!(m-t)!}=0 $$</div> <br></br></mdoxml> <mdoxml version="1.0"><p>Think about the binomial theorem.</p></mdoxml> 2 3 0 0 0 0 1 Prove that the sum from t=0 to m of (-1)^t/t!(m-t)! is zero. Advanced Algebra Binomial Theorem Using, Applying and Reasoning about Mathematics Mathematical reasoning & proof