How many different words can be formed by taking 4 letters from the word arrange without replacement?

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.

How many four-letter words with or without meaning, can be formed out of the letters of the word ‘LOGARITHMS’, if repetition of letters is not allowed?A) 40B) 400C) 5040D) 2520

Answer

Verified

Hint: We can take the letters in the given word and count them. Then we can find the permutation of forming 4 letters words with the letters of the given words by calculating the permutation of selecting 4 objects from n objects without replacement, where n is the number of letters in the given word which is obtained by the formula, ${}^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}$

Complete step by step solution:
We have the word ‘LOGARITHMS’.
We can count the letters. After counting, we can say that there are 10 letters in the given word.
$ \Rightarrow n = 10$
Now we need to form four letter words from these 10 numbers. As the words can be with or without meaning, we can take all the possible ways of arrangements.
As the repetition is not allowed, we can use the equation ${}^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}$ where n is the number of objects and r is the number of objects needed to be selected.
So, the number of four-letter words can be formed is given by,
$ \Rightarrow {}^{10}{P_4} = \dfrac{{10!}}{{\left( {10 - 4} \right)!}}$
So we have,
$ \Rightarrow {}^{10}{P_4} = \dfrac{{10!}}{{6!}}$
Using properties of factorial, we can write the numerator as,
$ \Rightarrow {}^{10}{P_4} = \dfrac{{10 \times 9 \times 8 \times 7 \times 6!}}{{6!}}$
On cancelling common terms we get,
$ \Rightarrow {}^{10}{P_4} = 10 \times 9 \times 8 \times 7$
Hence we have,
$ \Rightarrow {}^{10}{P_4} = 5040$
Therefore, the number of four-letter words that can be formed is 5040.

So the correct answer is option C.

Note: Alternate method to solve this problem is by,
We have 10 letters that have to be arranged in four places. It is given that repetition of letters is not allowed. So the letter once used cannot be used again.
So, in the $1^{\text{st}}$ place, we place any of the 10 letters. In the second place we can put any of the remaining 9 letters. In $3^{\text{rd}}$ place we can have any of the 8 letters and in the last place any of the remaining 7 letters can be used.
So, the total arrangement is given by, $10 \times 9 \times 8 \times 7 = 5040$ .
Therefore, the number of words that can be formed is 5040.

How many different ways can 4 letters be arranged?

The answer is 4! = 24. The first space can be filled by any one of the four letters.

How many different words can be formed by taking 4 letters out of the letters of word again with each word has to start with a?

Now total number of ways: 1680+18+756=2454.

How many different 4 letter words can be formed form the letters in the word math?

Therefore, the number of ways in which four letters of the word MATHEMATICS can be arranged is 2454.

How many different words of 4 letters can be formed with the letters of the word examination?

2454 different permutations can be formed from the letter of the word EXAMINATION taken four at a time.