Math Problem Statement

  P(7, 4) · P(6, 3)

P(5, 4)

Solution

The expression you've provided involves permutations. The permutation formula is defined as:

P(n,r)=n!(nr)!P(n, r) = \frac{n!}{(n - r)!}

where:

  • nn is the total number of items.
  • rr is the number of items to be chosen.

Let's break down each part of the expression:

1. P(7,4)P(7, 4)

P(7,4)=7!(74)!=7!3!P(7, 4) = \frac{7!}{(7 - 4)!} = \frac{7!}{3!}

7!=7×6×5×4×3×2×1=50407! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040

3!=3×2×1=63! = 3 \times 2 \times 1 = 6

P(7,4)=50406=840P(7, 4) = \frac{5040}{6} = 840

2. P(6,3)P(6, 3)

P(6,3)=6!(63)!=6!3!P(6, 3) = \frac{6!}{(6 - 3)!} = \frac{6!}{3!}

6!=6×5×4×3×2×1=7206! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720

P(6,3)=7206=120P(6, 3) = \frac{720}{6} = 120

3. P(5,4)P(5, 4)

P(5,4)=5!(54)!=5!1!P(5, 4) = \frac{5!}{(5 - 4)!} = \frac{5!}{1!}

5!=5×4×3×2×1=1205! = 5 \times 4 \times 3 \times 2 \times 1 = 120

1!=11! = 1

P(5,4)=1201=120P(5, 4) = \frac{120}{1} = 120

Now, combine them as per the given expression:

P(7,4)×P(6,3)=840×120=100800P(7, 4) \times P(6, 3) = 840 \times 120 = 100800

100800P(5,4)=100800120=840\frac{100800}{P(5, 4)} = \frac{100800}{120} = 840

So, the final value of the expression P(7,4)P(6,3)/P(5,4)P(7, 4) \cdot P(6, 3) / P(5, 4) is 840.

Would you like further details or explanations? Here are five related questions for further practice:

  1. What is P(8,5)P(8, 5) and how does it compare to P(7,4)P(7, 4)?
  2. How would you simplify P(7,4)P(6,3)P(5,2)P(7, 4) \cdot P(6, 3) \cdot P(5, 2)?
  3. How does P(n,r)P(n, r) change if you increase nn by 1 but keep rr the same?
  4. How can you express a combination C(n,r)C(n, r) in terms of permutations?
  5. How do permutations differ from combinations in terms of order and selection?

Tip: Remember that permutations consider the order of selection, making them useful in scenarios where the sequence matters.

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Math Problem Analysis

Mathematical Concepts

Permutations

Formulas

Permutation formula: P(n, r) = n! / (n - r)!

Theorems

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Suitable Grade Level

Advanced High School