Question

Jayden is working with the set of numbers 0, 1, 2, 4, 8, 16. By adding 2 or more different numbers from this set, Jayden can make different sums. A few examples are: 0+8=8, 1+4+8=13 , and 8+16=24. He cannot repeat a number in any one sum. How many different sums are possible?

Answer 1

The previous provided solution is not correct. Since, the order doesn't matter when you add (commutative property if you want to be fancy), this is a combination problem, not a permutation problem (if you are not familiar with these terms, see solution #2 below)

We have 6 total numbers.

How many sums can be made picking out any 2 numbers? 6C2 = 15 (same problem as "total number of handshakes in a company of 6 people)

Total possible sums by picking out any 3 numbers: 6C3 = 20

4 numbers: 6C4 = 15

5 numbers: 6C5 = 6

6 numbers: 6C6 = 1

So, the answer is 15 + 20 + 15 + 6 + 1 = 57

Solution #2 (for those in earlier grades who don't know how to calculate 6c3 yet)

You either use a number or you don't: that's 2 options for each number.

So, this a fundamental counting problem

2 × 2 × 2 × 2 × 2 × 2 =
`2^6` = 64

The condition of the problem is " By adding 2 or more different numbers from this set"

So we we have to take away:

when we don't use any of the numbers, "no" for each number:

0 1 2 4 8 16

no no no no no no (that's 6C0 btw in solution #1)

1 total sum

when we don't use each number once. That's 6 obviously.

6 total different sums

So we we have to take away: (6+1 ) from 64

Answer: 57

Answer 2

There are 1956 different sums possible using the set of numbers 0, 1, 2, 4, 8, and 16.

Step-by-step explanation:

To find the number of different sums that can be made using the set of numbers 0, 1, 2, 4, 8, 16, we can use a method called combinations. We can start with the smallest sum possible, which is 0. Then, we can add one number at a time to form new sums.

So, there are 6 possible sums with just one number. Next, we can add two numbers together. There are 6 possibilities for the first number and 5 possibilities for the second number, giving us a total of 6 x 5 = 30 possible sums. Similarly, we can continue adding three, four, five, and six numbers together.

Adding three numbers gives us 6 x 5 x 4 = 120 possible sums. Adding four numbers gives us 6 x 5 x 4 x 3 = 360 possible sums. Adding five numbers gives us 6 x 5 x 4 x 3 x 2 = 720 possible sums. And finally, adding all six numbers gives us 6 x 5 x 4 x 3 x 2 x 1 = 720 possible sums.

Therefore, the total number of different sums that can be made is 6 + 30 + 120 + 360 + 720 + 720 = 1956.