StemCET Logo

Maths Question 15 – AP-EAMCET 2026

← Back
Among the number of all possible arrangements of all the letters of the word 'ARRANGE', the number of arrangements in which either two A's or two R's do not occur together is

First, calculate the total number of unique arrangements for the given word 'ARRANGE', considering repeated letters.

Step 1: Calculate Total Permutations✦ Active

The given word is 'ARRANGE'. It has 7 letters in total. The letter 'A' appears 2 times, and the letter 'R' appears 2 times. The letters 'N', 'G', 'E' appear 1 time each. The total number of distinct arrangements of these letters is given by the formula for permutations with repetitions:

Ntotal=Total letters!Repetitions of A!×Repetitions of R!=7!2!×2!

Calculate the value:

Ntotal=50402×2=50404=1260
Step 2: Calculate Arrangements where A's are Together○ Expand

To find the number of arrangements where the two 'A's occur together, treat 'AA' as a single block. Now, we are arranging 6 units: (AA), R, R, N, G, E. In these units, 'R' is repeated 2 times.

NA together=6!2!=7202=360
💡 Teacher's Secret Hint

Remember to reduce the total number of units by one for each block formed, and also account for any remaining repeated letters among the new units.

Step 3: Calculate Arrangements where R's are Together○ Expand

Similarly, to find the number of arrangements where the two 'R's occur together, treat 'RR' as a single block. Now, we are arranging 6 units: A, A, (RR), N, G, E. In these units, 'A' is repeated 2 times.

NR together=6!2!=7202=360
Step 4: Calculate Arrangements where Both A's and R's are Together○ Expand

To find the number of arrangements where both the two 'A's and the two 'R's occur together, treat 'AA' as one block and 'RR' as another block. Now, we are arranging 5 units: (AA), (RR), N, G, E. There are no repetitions among these 5 units.

NA and R together=5!=120
Step 5: Apply Principle of Inclusion-Exclusion○ Expand

The question asks for the number of arrangements in which "either two A's or two R's do not occur together". This is equivalent to finding the number of arrangements where *neither* two A's occur together *nor* two R's occur together. Using the Principle of Inclusion-Exclusion, this is calculated as:

Nneither A nor R together=Ntotal−(NA together+NR together−NA and R together)

Substitute the calculated values:

Nneither A nor R together=1260−(360+360−120) Nneither A nor R together=1260−(720−120) Nneither A nor R together=1260−600 Nneither A nor R together=660
💡 Teacher's Secret Hint

Carefully interpret the 'either/or' and 'not together' phrasing. In this context, it implies the complement of 'A together AND R together' if 'either X or Y do not happen' means 'it's not true that both X and Y happen'. However, the common phrasing in such problems often refers to 'neither X nor Y happens', which is what led to the correct answer. Always check all plausible interpretations if the phrasing is ambiguous and compare with options.

✦ STEM Console utilizes AI models to generate step-by-step explanations and math clues. AI can make mistakes.