Studdy AI Logo
Math

Question Find an expression for the number of permutations of 3 items from nn items without using factorial notation.

Studdy Solution

STEP 1

Assumptions
1. nP3\mathrm{nP} 3 represents the number of permutations of nn distinct objects taken 3 at a time.
2. Factorial notation is typically defined as n!=n×(n−1)×(n−2)×…×2×1n! = n \times (n-1) \times (n-2) \times \ldots \times 2 \times 1.
3. The permutation formula is given by nPr=n!(n−r)!\mathrm{nP} r = \frac{n!}{(n-r)!} where nn is the total number of objects and rr is the number of objects to be arranged.

STEP 2

Write down the permutation formula for nP3\mathrm{nP} 3 using the factorial notation.
nP3=n!(n−3)!\mathrm{nP} 3 = \frac{n!}{(n-3)!}

STEP 3

Expand the factorial in the numerator n!n! up to (n−3)!(n-3)!.
n!=n×(n−1)×(n−2)×(n−3)!n! = n \times (n-1) \times (n-2) \times (n-3)!

STEP 4

Substitute the expanded form of n!n! into the permutation formula.
nP3=n×(n−1)×(n−2)×(n−3)!(n−3)!\mathrm{nP} 3 = \frac{n \times (n-1) \times (n-2) \times (n-3)!}{(n-3)!}

STEP 5

Cancel out the common factorial terms in the numerator and the denominator.
nP3=n×(n−1)×(n−2)\mathrm{nP} 3 = n \times (n-1) \times (n-2)

STEP 6

The expression n×(n−1)×(n−2)n \times (n-1) \times (n-2) is the equivalent expression to nP3\mathrm{nP} 3 without using factorial notation.
The equivalent expression to nP3\mathrm{nP} 3 with no factorial notation is n×(n−1)×(n−2)n \times (n-1) \times (n-2).

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord