Exercise:
Solve the following equation n∑k=1(n−k)(nk)=441
Solution: we have n∑k=1(n−k)(nk)=441⇔n∑k=1n(nk)−n∑k=1k(nk)=441
We know that from Binomial theorem (x+y)n=n∑k=0(nk)xn−kyk where k≤n
Put x=1&y=1 to get n∑k=0(nk)=2n⇔(n0)+n∑k=1=2n where (n0)=1
So n∑k=1(nk)=2n−1 thus n∑k=1n(nk)=nn∑k=1(nk)=n(2n−1)=n2n−n....(∗)
Notice that, (nk)=n!(n−k)!k!=n(n−1)!k(k−1)!(((n−1)−(k−1))!=nk(n−1k−1)
hence k(nk)=n(n−1k−1)⇔n∑k=1k(nk)=n−1∑k=0n(n−1k−1)=n2n−1....(∗∗)
Thus n∑k=1n(nk)−n∑k=1k(nk)=n2n−n−n2n−1=441
⇒n2n−n−n2n2=441⇒2n(n−n2)=441+n⇔2n(n2)=441+n
⇒n2n=882+2n⇒n(2n−2)=882
By using numerical method we get n=7
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