Question

For each of the subsets of set $\{1,2,3,4,5,6,...,n\}$; let G be the greatest number. in that set
Then, find the sum of all such Gs.

• A. $n{.2}^n-2^{n-1}+1$
• B. ${1.2}^0+{2.2}^1+{3.2}^2+....+(n-1)2^{n-2}+n{.2}^{n-1}$
• C. $n{.2}^n-2^n+1$
• D. $n{.2}^n+2^{n-1}+1$

Answer 1

Answer: (B), (C)

${1.2}^0+{2.2}^1+{3.2}^2+....+(n-1)2^{n-2}+n{.2}^{n-1}$
$n{.2}^n-2^n+1$

Since ${}^{\prime }{n}^{\prime }$ is the largest number in the whole set, every subset in which ${}^{\prime }{n}^{\prime }$ is present will have $G=n$
We have $2^{n-1}$ subsets which contain ${}^{\prime }{n}^{\prime }$.
(Fixing ${}^{\prime }{n}^{\prime }$, the other $(n-1)$ numbers can be included or not included).
Now we count the number of subsets in which $(n-1)$ appears, but ${}^{\prime }{n}^{\prime }$ does not appear.
Using a similar argument, we find that there are $2^{n-2}$ such subsets.
We continue our argument repeatedly and find the sum of all $G^{\prime }s$;
$n{.2}^{n-1}+(n-1)2^{n-2}+(n-2)2^{n-3}+....+{2.2}^2+{2.2}^0$
Let $S={1.2}^0+{2.2}^1+{3.2}^2+....+(n-1)2^{n-2}+n{.2}^{n-1}$
$S=n.2^n-2^n+1$