What is the number of ways in which 3 holiday travel tickets are to be given to 10 employees of an organization, if each employee is eligible for any one or more of the tickets?
1000
Calculating the Number of Ways to Distribute Holiday Travel Tickets
The question asks us to find the number of ways to distribute 3 distinct holiday travel tickets among 10 employees. A key condition mentioned is that each employee is eligible for any one or more of the tickets. This means an employee can receive zero tickets, one ticket, or multiple tickets (up to all three).
This is a problem involving the distribution of distinct items (the tickets) into distinct recipients (the employees), where repetition is allowed for the recipients (an employee can receive multiple tickets). We can think of this from the perspective of each ticket.
- For the first holiday travel ticket, there are 10 possible employees it can be given to.
- For the second holiday travel ticket, since an employee can receive more than one ticket, there are still 10 possible employees it can be given to.
- For the third holiday travel ticket, similarly, there are 10 possible employees it can be given to.
Since the choice for each ticket is independent of the choices for the other tickets, we can find the total number of ways by multiplying the number of options for each ticket.
Total number of ways = (Number of choices for Ticket 1) × (Number of choices for Ticket 2) × (Number of choices for Ticket 3)
Total number of ways = $10 \times 10 \times 10$
Total number of ways = $10^3$
Total number of ways = 1000
Therefore, there are 1000 ways to distribute the 3 holiday travel tickets among the 10 employees, given that each employee can receive any number of tickets.
Understanding Distribution Problems in Combinatorics
Problems like this often fall under the category of permutations with repetition. When we distribute $k$ distinct items among $n$ distinct recipients where each recipient can receive any number of items, the total number of ways is $n^k$. In this specific problem:
- Number of distinct recipients (employees), $n = 10$.
- Number of distinct items (tickets), $k = 3$.
Using the formula, the number of ways is $10^3 = 1000$.
| Ticket | Number of Employee Choices |
|---|---|
| Ticket 1 | 10 |
| Ticket 2 | 10 |
| Ticket 3 | 10 |
Multiplying the choices together gives the total number of ways.
Revision Table: Key Concepts
| Concept | Description | Formula Example |
|---|---|---|
| Permutation | Arrangement of distinct items where order matters. | Permutations of $n$ items taken $r$ at a time: $P(n, r) = \frac{n!}{(n-r)!}$ |
| Combination | Selection of items where order does not matter. | Combinations of $n$ items taken $r$ at a time: $C(n, r) = \frac{n!}{r!(n-r)!}$ |
| Permutation with Repetition | Arrangement where items can be repeated. Also applies to distributing distinct items to distinct recipients with repetition. | Number of ways to arrange $k$ items from $n$ with repetition: $n^k$ |
Additional Information on Ticket Distribution Problems
It's important to distinguish this problem from others in combinatorics:
- If tickets were identical: If the 3 tickets were identical, and each employee could get any number, this would be a stars and bars problem ($\binom{n+k-1}{k}$ or $\binom{n+k-1}{n-1}$). But the tickets are distinct holiday travel tickets.
- If employees could get at most one ticket: If each of the 10 employees could get at most one ticket (and the tickets are distinct), this would be a permutation problem $P(10, 3)$, selecting and arranging 3 employees out of 10 to receive the distinct tickets. However, the problem states each employee is eligible for "any one or more", implying no restriction on receiving multiple tickets.
Based on the problem statement, the items (tickets) are distinct, the recipients (employees) are distinct, and repetition (an employee getting multiple tickets) is allowed. This confirms the $n^k$ approach is correct, with $n=10$ and $k=3$.
Important Questions from Permutations and Combinations
What is the number of 6-digit numbers that can be formed only by using 0, 1, 2, 3, 4 and 5 (each once); and divisible by 6 ?
Consider the following statements for a fixed natural number n:
1. C(n, r) is greatest if n = 2r
2. C(n, r) is greatest if n = 2r - 1 and n = 2r + 1
Which of the statements given above is/are correct ?
Let x be the number of permutations of the word ‘PERMUTATIONS’ and y be the number of permutations of the word ‘COMBINATIONS’. Which one of the following is correct ?
A polygon has 44 diagonals then the number of its sides is
The number of ways in which 3-holiday tickets can be given to 20 employees of an organization if each employee is eligible for any one or more of the tickets, is
