Question

Slips of paper numbers 1 through 14 are placed in a hat. In how many ways can you draw two numbers with replacement that total 12?

Answer 1

Hint: In the given question, we have been asked to find the possible number of ways we can draw two numbers whose sum is 12. In order to solve this question, we assume two numbers are ‘x’ and ‘y’. Then we need to write all the possible values of ‘x’ and after that we have to write all the possible values of ‘y’ corresponding to the value of ‘x’ in order to satisfy the required condition i.e. x + y = 12.

Complete step by step solution:
Let assume,
The number you draw first is ‘x’ and the number you draw second be ‘y’.
Now, according to question,
\[\Rightarrow x+y=12\]
According to the above equation, we can say that\[x\le 12\].
So assume that we have following numbers for the first draw, i.e.
\[\Rightarrow x\in \left\{ 1,2,3,4,5,67,8,9,10,11 \right\}\]
Now, assuming the value of ‘y’ corresponding to the value of ‘x’,
For example is x = 11 then, y = 1 in order to have x + y = 12
Therefore,
Possible values of ‘y’ are,
\[\Rightarrow y\in \left\{ 11,10,9,8,7,6,5,4,3,2,1 \right\}\]
Thus, we have total to 11 possible value of ‘x’ and each have exactly one value for ‘y’ in order to satisfy the required condition, i.e. x + y = 12
Therefore,
All the possible ways are;
1.(x, y) = (1, 11)
2.(x, y) = (2, 10)
3.(x, y) = (3, 9)
4.(x, y) = (4, 8)
5.(x, y) = (5, 7)
6.(x, y) = (6, 6)
7.(x, y) = (7, 5)
8.(x, y) = (8, 4)
9.(x, y) = (9, 3)
10.(x, y) = (10, 2)
11.(x, y) = (11, 1)

\[\therefore \] There are 11 possible ways, we can draw two numbers with replacement that total 12.

Note: To answer these types of questions, students should assume the two numbers and required to find out all the possible values associated with that number. While thinking of the number that belongs to a specific variable, students should always keep in mind the necessary condition that is given in the question for example; here in this question the required condition while making the pair we have to keep in mind that x + y should be equal to 12.