# comb or perm?

• Aug 2nd 2010, 07:27 AM
sfspitfire23
comb or perm?
From the set of numbers: {1, 2, 3, 4, 5, 6}, how many different sums can be formed by summing up any two numbers in the set?

I'm thinking this could be a permutation because 1+2 and 2+1 are not two different sums. So,
$\displaystyle 6P2$=30.

Is this correct? Or would it be a combination...
• Aug 2nd 2010, 07:30 AM
Quote:

Originally Posted by sfspitfire23
From the set of numbers: {1, 2, 3, 4, 5, 6}, how many different sums can be formed by summing up any two numbers in the set?

I'm thinking this could be a permutation because 1+2 and 2+1 are not two different sums. So,
$\displaystyle 6P2$=30.

Is this correct? Or would it be a combination...

You counted the number of ways to arrange sets of two digits (permutations), which include 1,2 and 2,1.

Instead, since 1+2=2+1, you should count the number of combinations (or selections).
• Aug 2nd 2010, 02:30 PM
awkward
You should count neither permutations nor combinations, because some distinct combinations give the same sum. For example, 2+4 = 1+5.

So it's not a permutation or combination problem, it's a "think about it and figure out the answer" problem.
• Aug 2nd 2010, 03:08 PM
Yes,
count combinations if you want the sums of different numbers,
but not if you want different answers!

they will equal the sum of the numbers that "flank" them,

ie 2+3=1+4, 3+4=2+5, 4+5=3+6

if a number has 2 numbers to the left and 2 numbers to the right,
you get (flanking 3)...2+4=5+1, (flanking 4)...3+5=6+2.

Flanking both 3 and 4...2+5=6+1
• Aug 2nd 2010, 03:49 PM
Plato
Quote:

Originally Posted by sfspitfire23
[B]From the set of numbers: {1, 2, 3, 4, 5, 6}, how many different sums can be formed by summing up any two numbers in the set?
would it be a combination...

Again, as has been pointed out, it is neither as combination nor permutation problem.
If we add two of those numbers the minimum sum is 3 and the maximum is 11.
So how many different sums are there?
• Aug 2nd 2010, 08:52 PM
Soroban
Hello, sfspitfire23!

Quote:

From the set of numbers: {1, 2, 3, 4, 5, 6},
how many different sums can be formed by adding any two numbers in the set?

Assuming numbers can be repeated, we have this addition table:

. . $\displaystyle \begin{array}{c|cccccc} + & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ 5 & 6 & 7 & 8 & 9 & 10 & 11 \\ 6 & 7 & 8 & 9 & 10 & 11 & 12 \end{array}$

There are eleven different sums (2 through 12).

If numbers can not be repeated, we have this table:

. . $\displaystyle \begin{array}{c|cccccc} + & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline 1 & - & 3 & 4 & 5 & 6 & 7 \\ 2 & 3 & - & 5 & 6 & 7 & 8 \\ 3 & 4 & 5 & - & 7 & 8 & 9 \\ 4 & 5 & 6 & 7 & - & 9 & 10 \\ 5 & 6 & 7 & 8 & 9 & - & 11 \\ 6 & 7 & 8 & 9 & 10 & 11 & - \end{array}$

There are nine different sums (3 through 11).

• Aug 3rd 2010, 04:23 AM
$\displaystyle \binom{6}{2}-6$