Question

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

Answer 1

To solve this question, we have to determine the possibility of outcomes of 5 in dices. So, we have to form cases according to how many dice shows 5 and also which dice shows 5.

Complete step-by-step answer:
Now, there are three possibilities. One is that, only one dice shows 5. Another is that, any two of the dice show 5. And the last possibility is that all three of the dice shows 5. So, we are going to form cases according to them.
Case I: - When all the three dice show five, there is only one way of doing this, so the total number of ways of doing so is 1.
Case II: - When any two of the dice shows five and the other dice shows any number other than 5, the number of ways to do so is obtained by selecting in which dice, 5 is coming. So, the number of ways of selecting the dice is ${}^{3}{{C}_{2}}$ . The number of ways in which 5 can come in these dice are one. Now, the third dice will have 5 choices (excluding five). So, the total number of ways of doing so such that case II is satisfied is given by $={}^{3}{{C}_{2}}\times 1\times 5=15$
Case III: - When any one of the dice shows 5 and the rest two of the dice show numbers other than5, the number of ways to do so is obtained by first selecting in which dice 5 is coming. So, the number of ways of selecting the dice is ${}^{3}{{C}_{1}}$ . The number of ways in which 5 can come in this dice is 1.
Now for the other two dice, they have 5 choices each.
So, the total number of ways of doing so such that the case III is satisfies is given by $={}^{3}{{C}_{1}}\times 1\times 5\times 5=75$
Now, to obtain the overall number of ways, we will add three cases. After adding, we get
Total ways $=1+15+75$
$=91$

Note: The alternate method of doing is by finding the total number of possible outcomes and total number of outcomes in which no five is there. So,
Total number of outcomes $=6\times 6\times 6=216$
Total number of outcomes in which no five is obtained $=5\times 5\times 5=125$
After subtracting them, we will get the total number of ways in which there is at least one five. After subtracting, we get: -
Total ways $=216-125=91$