If Quantity A is the number of ways to assign a number from 1 to 5 without repetition to each of four people, and Quantity B is the number of ways to assign a number from 1 to 5 without repetition to each of 5 people, then which of the following statements is correct with respect to Quantities A and B?
Understanding Assignment Problems with Permutations
This question involves assigning unique items (numbers from 1 to 5) to a set of distinct positions (people) without repeating the items. This type of problem is addressed using permutations, as the order in which the numbers are assigned to different people matters.
The formula for calculating the number of permutations of selecting and arranging \(k\) items from a set of \(n\) distinct items is given by: $$P(n, k) = \frac{n!}{(n-k)!}$$ where \(n!\) (n factorial) is the product of all positive integers up to \(n\).
Calculating Quantity A: Assigning to Four People
Quantity A is defined as the number of ways to assign a number from 1 to 5 without repetition to each of four people.
- Total number of available numbers (items), \(n = 5\).
- Number of people (positions to fill), \(k = 4\).
Using the permutation formula \(P(n, k)\):
$$ \text{Quantity A} = P(5, 4) = \frac{5!}{(5-4)!} $$ $$ P(5, 4) = \frac{5!}{1!} $$ $$ P(5, 4) = \frac{5 \times 4 \times 3 \times 2 \times 1}{1} $$ $$ P(5, 4) = \frac{120}{1} $$ $$ \text{Quantity A} = 120 $$So, there are 120 ways to assign numbers from 1 to 5 without repetition to four people.
Calculating Quantity B: Assigning to Five People
Quantity B is defined as the number of ways to assign a number from 1 to 5 without repetition to each of 5 people.
- Total number of available numbers (items), \(n = 5\).
- Number of people (positions to fill), \(k = 5\).
Using the permutation formula \(P(n, k)\):
$$ \text{Quantity B} = P(5, 5) = \frac{5!}{(5-5)!} $$ $$ P(5, 5) = \frac{5!}{0!} $$By definition, \(0! = 1\).
$$ P(5, 5) = \frac{5!}{1} $$ $$ P(5, 5) = 5 \times 4 \times 3 \times 2 \times 1 $$ $$ \text{Quantity B} = 120 $$So, there are 120 ways to assign numbers from 1 to 5 without repetition to five people.
Comparing Quantity A and Quantity B
Now we compare the calculated values for Quantity A and Quantity B:
- Quantity A = 120
- Quantity B = 120
Since 120 is equal to 120, Quantity A and Quantity B are equal.
| Quantity | Description | Calculation | Value |
|---|---|---|---|
| Quantity A | Assign 1-5 without repetition to 4 people | \(P(5, 4) = \frac{5!}{(5-4)!} = 120\) | 120 |
| Quantity B | Assign 1-5 without repetition to 5 people | \(P(5, 5) = \frac{5!}{(5-5)!} = 120\) | 120 |
Based on the calculations, both quantities are equal.
Revision Table: Permutations Concepts
| Concept | Description | Formula |
|---|---|---|
| Permutation | Number of ways to arrange \(k\) items from a set of \(n\) items where order matters and without repetition. | \(P(n, k) = \frac{n!}{(n-k)!}\) |
| Factorial | Product of all positive integers up to a given integer \(n\). | \(n! = n \times (n-1) \times \dots \times 2 \times 1\) |
| Zero Factorial | Defined value used in permutation/combination formulas. | \(0! = 1\) |
Additional Information: Combinations vs. Permutations
It's important to distinguish between permutations and combinations when solving counting problems.
- Permutations: Used when the order of selection or arrangement matters. The problem of assigning numbers to specific people is a permutation because assigning number 1 to Person A and 2 to Person B is different from assigning number 2 to Person A and 1 to Person B.
- Combinations: Used when the order of selection does not matter. For example, selecting a committee of 3 people from a group of 10 is a combination problem, because the order in which you pick the people doesn't change the committee itself. The formula for combinations is \(C(n, k) = \frac{n!}{k!(n-k)!}\).
In this question, since distinct numbers are assigned to distinct people, and the assignment to each person is unique, it is a permutation problem.
Important Questions from Permutation and Combination
m parallel lines cut n parallel lines giving rise to 60 parallelograms. What is the value of (m + n) ?
5-digit numbers are formed using the digits 0, 1, 2, 4, 5 without repetition. What is the percentage of numbers which are greater than 50,000 ?
In a race, there are 4 members in a team. Each member has to cover 5 km one after another. If the total time taken is 30 minutes, then what would have been the average speed?
Which of the following muscles regulates the exit of food from the stomach into the small intestine?
The number of (a,b,c), where a,b,c are positive integers such that abc = 30, is
