How many arrangements can be formed with letters of word GANGAPUR ?

Question

Accepted Answer

The correct option is 1:
The problem requires us to calculate the number of distinct arrangements (permutations) that can be formed using all the letters of the word GANGAPUR. When calculating permutations for a word containing repeated letters, the formula used is $\frac{n\!}{n\_1\! n\_2\! \dots}$, where $n$ is the total number of letters, and $n\_1, n\_2, \dots$ are the counts of each repeated letter. The word GANGAPUR has a total of $n=8$ letters. By inspection, we find that the letter 'G' is repeated 2 times, and the letter 'A' is also repeated 2 times. The other letters (N, P, U, R) are unique. Applying the formula, the total number of arrangements is $\frac{8\!}{2\! \times 2\!}$. Calculating the factorials gives $8\! = 40,320$ and $2\! \times 2\! = 4$. Therefore, the total number of arrangements is $\frac{40,320}{4} = 10,080$.

The incorrect option is 2:
The value 10,800 is not the correct result from the standard permutation formula for GANGAPUR. This numerical figure is likely the result of an arithmetic error made by a student who correctly identified the formula and the repeated letters. For instance, the numerator $8\! = 40,320$ must be divided by $4$. If a student incorrectly divided by a number close to 3.73, they might arrive at a number near 10,800. This indicates an understanding of the concept but a flaw in the final calculation step.

The incorrect option is 3:
The figure 50,400 is too large to be the correct number of arrangements for a word with only 8 letters, which has a maximum of $8\! = 40,320$ total permutations. Even if the letters were all distinct, the number of arrangements would be $40,320$, and since there are repetitions, the actual count must be significantly lower than $40,320$. This answer suggests a serious error, perhaps the student used an incorrect factorial or simply selected a high-value option. This option is $10 \times 7\!$, which has no relevance to the problem of arranging 8 letters with repetitions.

The incorrect option is 4:
The value 5,400 is far too small for the number of arrangements for a word with 8 letters. If the letters were all distinct, the arrangements would be 40,320, and even after dividing by the repeated factors, the result should be 10,080. A result of 5,400 suggests a fundamental error, such as mistaking the total number of letters for 7 (since $7\! = 5040$) or incorrectly using the formula for combinations instead of permutations. The correct formula clearly dictates a much larger result than 5,400.