Question

There are 5 floors and 3 guards. Each guard can get duty at any of the floors and all 3 can also be posted on the same floor. They are given duty on floors at random. The probability that no two guards are placed on the same floor is
1) \[\dfrac{{12}}{{25}}\]
2) \[\dfrac{1}{2}\]
3) \[\dfrac{{20}}{{81}}\]
4) \[\dfrac{3}{5}\]

Answer 1

Hint: We will find the probability by using the concept of the combination. As we need to find the probability that no two guards can be on the same floor which means that there are 5 floors so each guard has 5 possibilities and since there are 3 guards so the combination can be multiplied by the factorial of 3. Now, the formula for probability says that the number of outcomes is divided by total outcomes. So, we can find the total outcomes by multiplying three times.

Complete step-by-step answer:
Consider the data given which says that there is a total of 5 floors and 3 guards are appointed. From any 3 of the guards, they can do the duty on any of the 5 floors but not on the same floor. The duties are given to them randomly.
We will use the definition of probability which states that the number of outcomes is divided by a total number of outcomes to determine the probability of any event.
So, here we will find the number of outcomes first,
Since there are 5 floors and 3 guards then the number of combinations is given by \[{}^5{C_3}\].
Next, we know that the guards are set up randomly so the number of combinations found above gets multiplied by the factorial of 3.
Also, we can open the value of \[{}^n{C_r}\] as \[{}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}\]
Thus, the possible number of outcomes are as follows:
\[
  {}^5{C_3} \times 3! = \dfrac{{5!}}{{3!\left( {5 - 3} \right)!}} \times 3! \\
   = \dfrac{{5 \times 4 \times 3 \times 2!}}{{2!}} \\
   = 60 \\
\]
Thus, the possible number of outcomes is given by 60 ways.
Next, we will find the total number of outcomes,
Thus, we get,
\[ \Rightarrow 5 \times 5 \times 5 = 125\]
Hence, the total number of outcomes are 125 ways.
So, the probability that no two guards can be placed on the same floor is given by:
\[ \Rightarrow \dfrac{{60}}{{125}} = \dfrac{{12}}{{25}}\]
Thus, we get the probability as \[\dfrac{{12}}{{25}}\].
Hence, option A is correct.

Note: We can also determine the number of possibilities as the first guard can choose any of the 5 floors and select one of them then the number of choices left for the second guard is 4 as he cannot repeat the floor so he has 4 choices and similarly the third guard can also not choose the same 2 floors which are already taken so the number of choices left for 3 guards is 3. Hence the number of possible ways is \[5 \times 4 \times 3 = 60\] and the total number of ways are \[5 \times 5 \times 5 = 125\]. Hence, we will get the same answer that is \[ \Rightarrow \dfrac{{60}}{{125}} = \dfrac{{12}}{{25}}\].