How many ways can 3 letters be arranged?

You can make three independent choices, one for each of the three letters. For each choice you have 26 options (the letters in the alphabet). So the total number of combinations is

$$ 26 \cdot 26 \cdot 26 = 26^3 = 17576. $$

If you want the letters to be unique, the calculation changes slightly. You still have 26 options for the first choice, but for the second choice there are now only 25 options available (all letters except the one you already chose), and for the third choice there are 24 options available (all letters except the two you already chose). So this gives you:

$$ 26 \cdot 25 \cdot 24 = 15600. $$

Distinguishable Ways to Arrange the Word USA
The below step by step work generated by the word permutations calculator shows how to find how many different ways can the letters of the word USA be arranged.

Objective:
Find how many distinguishable ways are there to order the letters in the word USA.Step by step workout:
step 1 Address the formula, input parameters and values to find how many ways are there to order the letters USA.
Formula:
nPr =n!/(n1! n2! . . . nr!)Input parameters and values:
Total number of letters in USA:
n = 3

Distinct subsets:
Subsets : U = 1; S = 1; A = 1;
Subsets' count:
n1(U) = 1, n2(S) = 1, n3(A) = 1

step 2 Apply the values extracted from the word USA in the (nPr) permutations equation
nPr = 3!/(1! 1! 1! )

= 1 x 2 x 3/{(1) (1) (1)}

= 6/1

= 6
nPr of word USA = 6

Hence,
The letters of the word USA can be arranged in 6 distinct ways.

Apart from the word USA, you may try different words with various lengths with or without repetition of letters to observe how it affects the nPr word permutation calculation to find how many ways the letters in the given word can be arranged.

This section covers permutations and combinations.

Arranging Objects

The number of ways of arranging n unlike objects in a line is n! (pronounced ‘n factorial’). n! = n × (n – 1) × (n – 2) ×…× 3 × 2 × 1

Example

How many different ways can the letters P, Q, R, S be arranged?

The answer is 4! = 24.

This is because there are four spaces to be filled: _, _, _, _

The first space can be filled by any one of the four letters. The second space can be filled by any of the remaining 3 letters. The third space can be filled by any of the 2 remaining letters and the final space must be filled by the one remaining letter. The total number of possible arrangements is therefore 4 × 3 × 2 × 1 = 4!

• The number of ways of arranging n objects, of which p of one type are alike, q of a second type are alike, r of a third type are alike, etc is:

n!        .
p! q! r! …

Example

In how many ways can the letters in the word: STATISTICS be arranged?

There are 3 S’s, 2 I’s and 3 T’s in this word, therefore, the number of ways of arranging the letters are:

10!=50 400
3! 2! 3!

Rings and Roundabouts

• The number of ways of arranging n unlike objects in a ring when clockwise and anticlockwise arrangements are different is (n – 1)!

When clockwise and anti-clockwise arrangements are the same, the number of ways is ½ (n – 1)!

Example

Ten people go to a party. How many different ways can they be seated?

Anti-clockwise and clockwise arrangements are the same. Therefore, the total number of ways is ½ (10-1)! = 181 440

Combinations

The number of ways of selecting r objects from n unlike objects is:

Example

There are 10 balls in a bag numbered from 1 to 10. Three balls are selected at random. How many different ways are there of selecting the three balls?

10C3 =10!=10 × 9 × 8= 120
             3! (10 – 3)!3 × 2 × 1

Permutations

A permutation is an ordered arrangement.

• The number of ordered arrangements of r objects taken from n unlike objects is:

nPr =       n!       .
          (n – r)!

Example

In the Match of the Day’s goal of the month competition, you had to pick the top 3 goals out of 10. Since the order is important, it is the permutation formula which we use.

10P3 =10!
            7!

= 720

There are therefore 720 different ways of picking the top three goals.

Probability

The above facts can be used to help solve problems in probability.

Example

In the National Lottery, 6 numbers are chosen from 49. You win if the 6 balls you pick match the six balls selected by the machine. What is the probability of winning the National Lottery?

The number of ways of choosing 6 numbers from 49 is 49C6 = 13 983 816 .

Therefore the probability of winning the lottery is 1/13983816 = 0.000 000 071 5 (3sf), which is about a 1 in 14 million chance.

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