Question

How many seven letter words can be formed using the letters of the word ‘ALABAMA’?
A. 210
B. 520
C. 225
D. 525

Answer 1

Hint: In the given question the total number of letters in the given word is 7 but the letter ‘A’ is repeating 4 times so we will select 7 out of 7 words and then we will divide it by 4 factorial due to this repetition of letters.

Complete step-by-step answer:
We have been asked to find the number of seven letter words that can be formed using the letters of the word ‘ALABAMA’.
In the given word, there are a total of seven letters 4A, B, L and M.
So to find the total number of 7 letter words possibly is equal to selection of 7 letters out of 7 letters and then we will divide it by factorial 4 since there are four ‘A’ which is repetitive.
Total number of 7 lettered words $=\dfrac{^{7}{{P}_{7}}}{4!}$
We know that $^{n}{{P}_{r}}=\dfrac{n!}{(n-r)!}$
$\Rightarrow \dfrac{^{7}{{P}_{7}}}{4!}=\dfrac{7!}{\dfrac{(7-7)!}{4!}}=\dfrac{7!}{0!\times 4!}$
Since we know that 0! Is equal to 1
$=\dfrac{7!}{1!\times 4!}=210$
Therefore, the total number of words that can be formed is equal to 210 and the correct answer is option A.

Note: Be careful while doing calculation especially while finding the value of $^{7}{{P}_{7}}$ . One must be aware not to use C instead of P while solving this question. If one uses C and applies the formula for combination, then one will get the incorrect answer. Also, remember that $\left( ^{n}{{P}_{r}} \right)$ means the number of permutations of n objects taken ‘r’ at a time.