Counting Score: 0/10 0/10 answered Question 2 Find the value of
990 40,320 39,916,800 165
The Correct Answer and Explanation is:
Correct answer: 990
When you see a symbol like 11P3^{11}P_{3}, read it as “the number of permutations of 11 objects taken 3 at a time.” A permutation is an ordered arrangement, so switching the order of the selected items gives a different outcome. That is why we multiply rather than simply count combinations.
Formula
For any non-negative integers nn and rr where r≤nr \le n, nPr = n!(n−r)!^{n}P_{r} \;=\; \frac{n!}{(n-r)!}
The factorial symbol “!” means “multiply all positive integers down to 1.”
So n!=n×(n−1)×(n−2)×⋯×1n! = n \times (n-1) \times (n-2) \times \dots \times 1.
Apply it to 11P3^{11}P_{3}
- Plug in the numbers: n=11n = 11 and r=3r = 3. 11P3 = 11!(11−3)! = 11!8!^{11}P_{3} \;=\; \frac{11!}{(11-3)!} \;=\; \frac{11!}{8!}
- Cancel the common part:
Notice that 11!=11×10×9×8!11! = 11 \times 10 \times 9 \times 8!.
The 8!8! terms in numerator and denominator cancel: 11×10×9×8!8!=11×10×9\frac{11 \times 10 \times 9 \times \cancel{8!}}{\cancel{8!}} = 11 \times 10 \times 9 - Multiply what’s left: 11×10=110,110×9=99011 \times 10 = 110,\quad 110 \times 9 = 990
Therefore, 11P3=990^{11}P_{3} = 990.
What it means
Imagine you have 11 distinct books and you want to arrange any 3 of them on a display shelf where position matters (first spot, second spot, third spot). There are 990 different ways to do that. If order did not matter—say you were merely choosing 3 books to take on a trip—then you’d use a combination 11C3^{11}C_{3} instead, which is smaller because repeated orders are merged into a single outcome.
Quick mental check
A rough sanity check helps avoid arithmetic slips:
- The number of ordered triples must be less than 11×11×11=1,33111 \times 11 \times 11 = 1{,}331 (all possible triples with repeats), because permutations forbid repeats.
- 990 lies comfortably below 1,331, while 40,320 and 39,916,800 are clearly far too large.
- 165 is more plausible but is actually the value of 11C3^{11}C_{3}, the combination counterpart.
That confirms 990 is both mathematically and intuitively consistent.
