# Three dice are rolled. Find the number of possible outcomes in which at least one die shows 5.

We have to find the number of total outcomes when three dices are rolled the outcomes can be different and can be the same, but one outcome of at least one dice is made fixed.

We will use the concept of multiplication because there are three sub jobs dependent on each other and one job is performed one after the other.

The choices of outcomes from one dice are six, there are three dices so choices for each dice will be six but the outcome of at least one dice is made fixed which means at most all dices would have a fixed outcome which is 5.

The number of outcomes which are formed when three dices are rolled one after the other but at least one dice is fixed at 5

= total number of possible outcomes – the number of possible outcomes in which 5 does not occur on any dice.

= 6 × 6 × 6 - 5 × 5 × 5 = 216 – 125 = 91

A number of possible outcomes in which 5 does not occur on any dice means that we have deducted 5 from all the six possible outcomes and now only five possible outcomes are there, which will give 125 total outcomes.

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