Question

How many seven-letter permutations can be formed from the letters of the word GIGGLES?
(a) 160
(b) 840
(c) 1200
(d) None of these

Answer 1

In this given problem, we are trying to form as many words as we can from the given word “GIGGLES”. So, to start with, we will check how many letters are repeating in the given word and then we also try to count the number of letters in that word. Once we find that we can write it in our given formula and thus by simplifying we will get our needed result.

Complete step by step solution:
According to the question, we are to find how many seven-letter permutations can be formed from the letters of the word GIGGLES.
Now, to start with, the answer is the same as that of the other contributions, but I would like to explain a little more how the calculations are obtained.
The number of ways an n-element set can be ordered is:
${{P}_{n}}=n!$
as long as the elements are different from each other.
But suppose the first element is repeated a times. Then, since all the repeats of the first element are indistinguishable from each other, in reality they will only have:
$\dfrac{n!}{a!}$
possible ordinations.
Finally, if the rest of the elements of the set can also be repeated a number of times each, i.e. we have b second elements, c third, etc., then we have:
${{P}_{n\left( a,b,c,... \right)}}=\dfrac{n!}{a!.b!.c!....}$
possibilities of ordering said set of elements.
Now, putting the values, we can see, G is repeating 3 times, and I,E,L and S are going once. And again, the word has 7 letters.
Then, the permutations we have are, $=\dfrac{7!}{3!\left( 1!\,\times 4 \right)}$
Now, again simplifying, we are getting, $=4\times 5\times 6\times 7=840$

So, the correct answer is “Option b”.

Note: In this solution, we are only considering with the given letters to us. But there also might be a case when we have to consider the vowels and consonants differently and count the number of words we can make out of it. That should be considered as another typical case of the solution.