How many 4 digit PINs are possible if digits can repeat?

If digits 0 through 9 can repeat, the number of possible 4 digit PINs is PR(10,4) = 10^4 = 10,000. Each position can hold any of the 10 digits independently.

What happens when n equals r in permutation with replacement?

When n equals r, the formula PR(n,r) = n^r gives n^n, which is always larger than n! (the result for a standard permutation without replacement). For example, PR(3,3) = 3^3 = 27, while P(3,3) = 3! = 6.

Permutation with Replacement Calculator | PR(n,r) = n^r

Permutation with Replacement Calculator

Find the number of ordered arrangements when elements can be reused. Uses the formula PR(n,r) = n^r.

PR(n,r) = n^r

Permutations with Replacement PR(n,r)

Formula: PR(n,r) = n^r — Order matters, replacement allowed

n — Number of Objects The size of the set to choose from

r — Sample Size How many elements to select

PR(n,r) Result

Step	Working	Result

PR(n,r) Reference Table

n (Objects)	r (Sample)	Formula	PR(n,r)
2	2	2²	4
3	2	3²	9
4	2	4²	16
4	3	4³	64
6	2	6²	36
6	4	6⁴	1,296
10	4	10⁴	10,000
26	3	26³	17,576
6	60	6⁶⁰	4.887 × 10⁴⁶

What Is Permutation with Replacement?

Permutation with replacement calculates how many ordered sequences you can form from a set of n distinct objects when you are allowed to reuse any object. This means the same element can appear more than once in your sequence, and the order in which elements appear matters.

The formula is straightforward: each position in the sequence of r elements can be filled by any of the n objects independently. So the total number of sequences is n multiplied by itself r times.

PR(n,r) = n^r

What the Variables Mean

n is the number of distinct objects in the set or population you are choosing from.

r is the number of elements in each sequence or sample you are forming.

Permutation with Replacement Examples

Permutation with replacement examples help illustrate how the formula applies in real situations.

Example 1: Letters from an Alphabet

How many 2 letter sequences can be formed from {a, b, c, d}? Here n = 4 and r = 2.

PR(4,2) = 4² = 16

The 16 sequences are: {a,a}, {a,b}, {a,c}, {a,d}, {b,a}, {b,b}, {b,c}, {b,d}, {c,a}, {c,b}, {c,c}, {c,d}, {d,a}, {d,b}, {d,c}, {d,d}.

Example 2: Rolling a Die

How many distinct sequences can result from rolling a standard 6 sided die 60 times? Here n = 6 and r = 60.

PR(6,60) = 6⁶⁰ = approximately 4.887 × 10⁴⁶

Example 3: 4 Digit PIN Codes

How many 4 digit PIN codes exist if each digit (0 through 9) can be reused? Here n = 10 and r = 4.

PR(10,4) = 10⁴ = 10,000

Permutation vs Permutation with Replacement

Permutation and permutation with replacement both count ordered arrangements, but they differ in whether items can be reused.

Feature	Permutation P(n,r)	Permutation with Replacement PR(n,r)
Order matters	Yes	Yes
Replacement allowed	No	Yes
Formula	n! / (n−r)!	n^r
Example P(4,2)	12	16
Requires n ≥ r	Yes	No

Because replacement is allowed, PR(n,r) is always greater than or equal to P(n,r) for the same values of n and r.

Scientific Notation for Large Results

Scientific notation handles very large permutation results that would otherwise be difficult to read. When n and r are both large, n^r grows exponentially and quickly exceeds billions or trillions. The calculator displays these in the form a × 10^b for easy reading. For example, PR(6,60) = 6⁶⁰ is shown as 4.887 × 10⁴⁶.

Frequently Asked Questions

The permutation with replacement formula is PR(n,r) = n^r, where n is the number of distinct objects in the set and r is the number of elements selected. Order matters and each element can be chosen more than once.

In a standard permutation, each element can only be chosen once and the formula is P(n,r) = n! / (n−r)!. In a permutation with replacement, elements can be reused and the formula is n^r. Permutations with replacement always produce a larger or equal count.

Use permutation with replacement when order matters and an item can be selected more than once. Common examples include creating PIN codes, rolling dice multiple times, and generating letter or number sequences where repetition is allowed.

If digits 0 through 9 can repeat, the number of possible 4 digit PINs is PR(10,4) = 10⁴ = 10,000. Each of the four positions can independently hold any of the 10 digits.

When n equals r, the formula gives nⁿ, which is always larger than n factorial (the result for a standard permutation). For example, PR(3,3) = 3³ = 27, while P(3,3) = 3! = 6.

Yes. Because replacement is allowed, r can be larger than n. For example, you can roll a 6 sided die (n = 6) 60 times (r = 60) and record all 60 results. Standard permutations without replacement require r to be less than or equal to n.