Question
Question: What is \[{}^8{P_3}\] ? What is \[{}^7{C_3}\] ? And what do they mean?...
What is 8P3 ? What is 7C3 ? And what do they mean?
Solution
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 mCn and its value is given by mCn=n!(m−n)!m!
And the number of arrangements of m objects taken n at a time is given by permutations and is represented as mPn and its value is given as mPn=(m−n)!m!
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 4C2 and its value is 4C2=2!(4−2)!4!=2!2!4!=2×1×2×14×3×2×1=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 5P2 and its value is 5P2=(5−2)!5!=3×2×15×4×3×2×1=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, 8P3 which means the number of ways of arranging eight things taken three at a time and is equal to 8P3=(8−3)!8!=5!8!
⇒8P3=5!8!=5×4×3×2×18×7×6×5×4×3×2×1=336
And 7C3 means the number of ways of selecting three things from seven things.
⇒7C3=3!(7−3)!7!=3!4!7!
⇒7C3=3!4!7!=3.2.1.4.3.2.17.6.5.4.3.2.1=35
Here 7C3 means on selecting the 3 things out of 7 things gives the value 35.
So, these are the meanings of 8P3 and 7C3 .
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.