Question

Ten different letters of the alphabet are given. Words with 4 letters are formed from these letters, then the number of words which have at least one letter repeated is
A.${10^4}$
B.$^{10}{P_4}$
C.$^{10}{C_4}$
D.4960

Answer 1

We find the total number of words formed using the given number of different letters and the length of word that has to be formed. Calculate the number of words formed which have no
repetition allowed using the method of permutation. Since the number of words formed having at least one letter repeated means that there can be more than one repetition, we subtract the number of words with no repetition from the total number of words formed.

• Permutation method gives us formula of choosing ‘r’ objects from total ‘n’ objects as $^n{P_r} = \dfrac{{n!}}{{(n - r)!}}$, where factorial opens up as $n! = n \times (n - 1)! = n \times (n - 1) \times (n- 2)!$ and so on.

Complete step-by-step answer:
We are given a total number of different letters available as 10.
Since we have to form a four letter word,
We can write the total number of four letter words formed as $10 \times 10 \times 10 \times 10$
$\Rightarrow$Total four letter words formed using 10 letters $= 10000$ ……….… (1)
Now we calculate the number of words formed which have no repetition allowed.
Since we have number of total letters as 10 and number of letters to choose as 4
$\Rightarrow n = 10,r = 4$
Number of 4 letter words without any repetition is given by$^{10}{P_4}$
Use the formula of combinations i.e. $^n{P_r} = \dfrac{{n!}}{{(n - r)!}}$
${ \Rightarrow ^{10}}{P_4} = \dfrac{{10!}}{{(10 - 4)!}}$
${ \Rightarrow ^{10}}{P_4} = \dfrac{{10!}}{{6!}}$
Use the formula of factorial to break the term in numerator.
${ \Rightarrow ^{10}}{P_4} = \dfrac{{10 \times 9 \times 8 \times 7 \times 6!}}{{6!}}$
Cancel the same terms from numerator and denominator.
${ \Rightarrow ^{10}}{P_4} = 10 \times 9 \times 8 \times 7$
${ \Rightarrow ^{10}}{P_4} = 5040$
$\Rightarrow$Number of words that have no repetition is 5040. ………..… (2)
We have to calculate the number of words that have at least one repeated letter.
Since we know at least one repeated letter means that it includes letters will repetition of one letter,
two letters, three letters and four letters.
$\Rightarrow$Number of words that have at least one letter repeated is given by subtracting
number of words with no repetition from total number of words formed
Subtract equation (2) from equation (1)
$\Rightarrow$Number of words with at least one repetition $= 10000 - 5040$
$\Rightarrow$Number of words with at least one repetition $= 4960$

$\therefore$Option D is correct.

Note: Students are likely to make the mistake of applying the combination formula while calculating the number of words with no repetition. Keep in mind we apply a combination formula where order doesn't matter, here we have to remove the repeated letter so the order does matter. Students might try solving all the cases of repetition separately but that is a long process, instead we find the words with no repetition and solve.