Question

What is \[{}^8{P_3}\] ? What is \[{}^7{C_3}\] ? And what do they mean?

Answer 1

Hint: We use the concepts of permutations and combinations to solve this problem. Permutations and combinations are all related to selecting and sorting or arranging things, in which we have to consider every single case to get a perfect value. We will also learn how to evaluate these permutations and combinations.

Complete step-by-step solution:
Firstly, consider that there are \[m\] different things or objects. And now, to select \[n\] things or objects from these, we use combinations, which gives us the number of ways of selecting \[n\] objects from \[m\] objects. It is represented as \[{}^m{C_n}\] and its value is given by \[{}^m{C_n} = \dfrac{{m!}}{{n!\left( {m - n} \right)!}}\]
And the number of arrangements of \[m\] objects taken \[n\] at a time is given by permutations and is represented as \[{}^m{P_n}\] and its value is given as \[{}^m{P_n} = \dfrac{{m!}}{{\left( {m - n} \right)!}}\]
For example, if we have four numbers say \[1,2,3,4\] , then number of ways of selecting two numbers out of these four is given by \[{}^4{C_2}\] and its value is \[{}^4{C_2} = \dfrac{{4!}}{{2!\left( {4 - 2} \right)!}} = \dfrac{{4!}}{{2!2!}} = \dfrac{{4 \times 3 \times 2 \times 1}}{{2 \times 1 \times 2 \times 1}} = 6\]
And the combinations are \[(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)\]
Take another example, in which there are five numbers say \[1,2,3,4,5\] and we need to arrange these five numbers when taken two at a time. And to do so, we use permutations and it is given by \[{}^5{P_2}\] and its value is \[{}^5{P_2} = \dfrac{{5!}}{{\left( {5 - 2} \right)!}} = \dfrac{{5 \times 4 \times 3 \times 2 \times 1}}{{3 \times 2 \times 1}} = 20\]
And the permutations are \[(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5),(5,1),(5,2),(5,3),(5,4),(4,1),(4,2),(4,3),(3,1),(3,2),(2,1)\]
Now, in the question, it is given that, \[{}^8{P_3}\] which means the number of ways of arranging eight things taken three at a time and is equal to \[{}^8{P_3} = \dfrac{{8!}}{{\left( {8 - 3} \right)!}} = \dfrac{{8!}}{{5!}}\]
\[ \Rightarrow {}^8{P_3} = \dfrac{{8!}}{{5!}} = \dfrac{{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}}{{5 \times 4 \times 3 \times 2 \times 1}} = 336\]
And \[{}^7{C_3}\] means the number of ways of selecting three things from seven things.
\[ \Rightarrow {}^7{C_3} = \dfrac{{7!}}{{3!\left( {7 - 3} \right)!}} = \dfrac{{7!}}{{3!4!}}\]
\[ \Rightarrow {}^7{C_3} = \dfrac{{7!}}{{3!4!}} = \dfrac{{7.6.5.4.3.2.1}}{{3.2.1.4.3.2.1}} = 35\]
Here ${}^7{C_3} $ means on selecting the 3 things out of 7 things gives the value 35.
So, these are the meanings of \[{}^8{P_3}\] and \[{}^7{C_3}\] .

Note: All the total things that are considered, are different things. And we get a positive integer as a result of permutations or combinations. If you get a negative value or a fractional value, then your solution has gone wrong in some way. In combinations, we just need to consider only a pair, but not its permutations. For example, the combinations of selecting two numbers from 1,2,3 are \[(1,2),(1,3),(2,3)\] . Here, we have to consider only one pair, i.e., either (1,2) or (2,1) but not both.