Permutation with Repetition Calculator (n to the power r)

Compute permutations with repetition n^r for choosing and arranging r items from n types. Also finds the number of possible PIN codes.

Input

Calculate the permutation with repetition n to the power r, the ways to arrange r items chosen from n types when repeats are allowed. Useful for counting PIN codes and passwords.

Number of types n

How many kinds you can choose from (integer, 0 or more)

Items to arrange r

Number of positions to fill (integer, 0 or more)

Result

n to the power r when n = 10 and r = 4

10,000

Number of digits

Ordinary permutation P(n, r)

5,040

Ratio to ordinary permutation

about 1.9841 times

Comparison

Allowing repeats gives about 1.9841 times as many arrangements as the ordinary permutation.

Quick reference table for n to the power r

Values for common combinations of n and r.

n and r	r = 1	r = 2	r = 3	r = 4	r = 5
n = 2	2	4	8	16	32
n = 4	4	16	64	256	1,024
n = 6	6	36	216	1,296	7,776
n = 8	8	64	512	4,096	32,768
n = 10	10	100	1,000	10,000	100,000

In a permutation with repetition each position has n independent choices, so the total is n multiplied by itself r times, n to the power r.

How it works

A permutation with repetition counts the ways to choose and arrange r items from n types when the same item may be used more than once and the order matters. Each of the r positions has n independent choices, so the total is n multiplied by itself r times, written n to the power r.
For example, a 4 digit PIN built from the 10 digits 0 through 9 has 10^4 = 10000 possibilities. Each position may be any of the ten digits and digits may repeat, which gives this value.
An ordinary permutation P(n, r) does not allow reuse, so it is always smaller than n^r. This tool compares the two and shows how many times larger the repeated permutation is.
When r is 0 there are no items to arrange, so there is exactly one empty arrangement and we define n^0 = 1.
For very large results, an approximate magnitude is also shown using a mantissa and a power of 10 in scientific notation.

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