Combinations Without Repetition: The Art of Choosing Without Repeating
Imagine you are a consumer with a fixed budget, standing in a market with five different fruits. You want to buy three of them, but you cannot buy the same fruit twice. How many different fruit baskets can you make? That is the essence of combinations without repetition — selecting a subset from a larger set where order does not matter and no item is used more than once.
The Everyday Intuition
Think of a committee being formed from a class of 30 students. The committee has 4 members. Whether you pick Ravi first and Priya second, or Priya first and Ravi second, the committee is the same. The order of selection is irrelevant. What matters is which 4 students are chosen, not the sequence in which they were picked. This is a combination — a selection where the arrangement inside the selection does not count.
Contrast this with a permutation, where order matters. If you were assigning four different roles (President, Vice-President, Secretary, Treasurer) to four students, then Ravi as President and Priya as Vice-President is a different outcome from Priya as President and Ravi as Vice-President. That is a permutation.
The Precise Meaning
A combination without repetition is a selection of r distinct objects from a set of n distinct objects, where the order of selection does not matter. The number of such combinations is given by:
(rn)=r!(n−r)!n!
Here:
- n is the total number of distinct objects available
- r is the number of objects you are selecting (0≤r≤n)
- n! (read "n factorial") means n×(n−1)×(n−2)×⋯×2×1
- (rn) is read as "n choose r"
The key condition: no repetition — once an object is chosen, it cannot be chosen again. This is why r cannot exceed n.
Why the Formula Works
The numerator n! counts all possible arrangements of all n objects. But we only care about r of them, and we do not care about the order of those r, nor about the order of the remaining n−r objects. So we divide by r! (to remove the ordering among the chosen ones) and by (n−r)! (to remove the ordering among the unchosen ones).
Why It Matters in Economics
Economics is full of situations where you need to count possibilities without repetition.
Consumer choice under a budget constraint. A consumer has a limited income and faces a set of goods. If the consumer can buy at most one unit of each good (indivisible goods, or "one per customer" offers), then the number of possible consumption bundles is a combination problem. If there are 10 goods and the consumer buys exactly 3 different ones, the number of possible bundles is (310).
Portfolio selection. An investor choosing 5 different stocks from a list of 20 to build a diversified portfolio. The order in which the stocks are bought does not matter — only which 5 stocks are held. The number of possible portfolios is (520).
Sampling without replacement. When a government agency surveys households without interviewing the same household twice, the number of possible samples of size r from a population of n households is (rn). This is the foundation of sampling theory in statistics, which is used extensively in economics for estimating national income, unemployment, and inflation.
In economics, combinations without repetition appear whenever you are selecting a set of distinct items — whether goods, stocks, households, or time periods — and the order of selection carries no economic meaning.
A Simple Example …