There are three distinct ways to add four positive odd numbers to obtain 10:
1 + 1 + 3 + 5 = 10
1 + 1 + 1 + 7 = 10
1 + 3 + 3 + 3 = 10
Here, distinct means that changing the order of the numbers on the left-hand side of an
equation does not count as a new solution. In how many distinct ways can we add eight odd
positive numbers to obtain 20? Justify your response. Hint: be systematic in your approach!
Perhaps there is a more elegant way, but the following should work.
Start with smallest to largest numbers:
1+1+1+1+1+1+1+13=20
1+1+1+1+1+1+3+11=20
1+1+1+1+1+1+5+9=20
1+1+1+1+1+1+7+7=20
1+1+1+1+1+3+3+9=20
1+1+1+1+1+3+5+7=20
1+1+1+1+1+5+5+5=20
1+1+1+1+3+....
keep going......
To find the number of distinct ways to add eight odd positive numbers to obtain 20, we can use a systematic approach.
Let's list out all the possible combinations of odd numbers that add up to 20 and check if they satisfy the conditions of distinctness.
We can start by visualizing the problem using stars and bars. We have 8 odd numbers (represented as stars) that need to be distributed among 20 'spaces' (represented as bars). For example, if we have the combination 1 + 1 + 1 + 1 + 3 + 3 + 5 + 5, we can represent it as:
* | * | * | * | | * | * | | * | * | * | * | | | * | * | | | | * | * |
Any combination can be represented in this way, with the number of spaces between each pair of bars indicating the value of the odd number.
Now, let's consider a systematic approach to find all possible combinations. We can create a table representing the number of odd numbers used for each value from 1 to 20.
Value | Count
-----------------
1 |
3 |
5 |
7 |
9 |
11 |
13 |
15 |
We need to fill in the count column in such a way that it satisfies the given conditions of distinctness.
Starting with the count values of 1, 3, 5, and 7, we can see that there is only 1 way to obtain these values, i.e., by using all 1's, 3's, 5's, or 7's.
For the value of 9, we can have two 1's and one 7, or one 1, one 3, and one 5. So, there are two distinct ways to obtain 9.
Similarly, for the value of 11, we can either have three 1's and one 7, or one 1, one 3, and one 7, or one 1 and two 5's. So, there are three distinct ways to obtain 11.
Continuing this process, we can fill in the counts for the remaining values:
Value | Count
------------------
1 | 1
3 | 1
5 | 1
7 | 1
9 | 2
11 | 3
13 | 5
15 | 6
Finally, we can sum up the counts to get the total number of distinct ways:
1 + 1 + 1 + 1 + 2 + 3 + 5 + 6 = 20
Therefore, there are 20 distinct ways to add eight odd positive numbers to obtain 20.