Introduction / Context:
The question asks for the value of a permutation expression, 100P2. This notation represents the number of ways to arrange 2 distinct items chosen from a set of 100 distinct items, where the order of the chosen items matters.
Given Data / Assumptions:
- The total number of distinct objects, n, is 100.
- The number of positions to be filled, r, is 2.
- We use the permutation formula nPr for arrangements where order is important.
Concept / Approach:
The formula for permutations is nPr = n * (n - 1) * (n - 2) ... up to r factors. For small r, this simplifies to a product of r consecutive integers starting from n and decreasing by 1 each time. For 100P2, we only need two factors: 100 and 99.
Step-by-Step Solution:
Step 1: Recall the formula for permutations.
nPr = n! / (n - r)!.
Step 2: Substitute n = 100 and r = 2.
100P2 = 100! / 98!.
Step 3: Cancel common terms in the factorials. 100! / 98! = 100 * 99 because all the factors from 98 downwards cancel out.
Step 4: Compute the product 100 * 99.
100 * 99 = 9900.
Verification / Alternative check:
You can also think of this as choosing the first position in 100 ways and the second position in 99 ways. Since each ordered pair is distinct, the total number of ordered pairs is 100 * 99, which again gives 9900.
Why Other Options Are Wrong:
10000 would correspond to 100 * 100, which ignores the fact that once one element is chosen, it cannot be chosen again. 8900 and 7900 are arbitrary numbers that do not arise from the correct permutation formula or any logical counting method here.
Common Pitfalls:
A frequent mistake is to mix up permutations and combinations and try to use nCr instead. Another error is to allow repetition, which would lead to 100^2, not the required product 100 * 99.
Final Answer:
The value of the permutation 100P2 is
9900.
Discussion & Comments