There are 7 boxes each written with letter to ( for the first box, for the 2nd and so on). You have 7 balls each with letter to ( for the first ball, for the 2nd and so on). You randomly put one ball in each box (blindfolded). Find the probability that you put 3 balls correctly (that is, you put the ball in the box with the same letter)?

Source: submitted by Jimel Mariano 9/25/2011

**SOLUTION**

We randomly put a ball in a box. We have 7 choices for the first box, then 6 for the second box, etc. The number of possible outcomes is .

**Putting 3 balls correctly**

We want to investigate the desirable outcomes of putting 3 balls correctly. Without loss of generality assume that we put 3 balls correctly in the 3 boxes labeled as shown in the diagram below

The remaining balls are . We cannot put ball in the fourth box because that would make 4 balls placed correctly instead of 3. But we can put one of the balls in the fourth box and ball somewhere else. We now consider the 3 cases depending on whether we place ball or in the fourth box.

**Case 1**:

Since ball is already in the wrong place, we can put any of the remaining balls in the fifth box. We have 3 choices

cannot be in sixth and cannot be in seventh

cannot be in seventh

cannot be in sixth

**Case 2**:

As long as we don’t put ball in the fifth box we will be OK. We have 3 choices

cannot be in seventh

no restriction on

**Case 3**:

As long as we don’t put ball in the fifth box we will be OK. We have 3 choices

cannot be in sixth

no restriction on

The number of desirable outcomes for putting 3 balls correctly equals 9.

**Probability of putting 3 balls correctly**

The same reasoning applies to any other three boxes. For example, the number of desirable outcomes for putting balls correctly in boxes labeled equals 9. How many ways can we choose 3 boxes from a set of seven? The answer is given by the “choose number”

Therefore the number of desirable outcomes for putting ANY 3 balls correctly equals

The probability of putting 3 balls correctly is

**Answer**: