JEE 2023 — Mathematics PYQ

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Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

If the letters of the word MATHS are permuted and all possible words so formed are arranged as in a dictionary with serial number, then the serial number of the word THAMS is:

Choose the correct answer:

Explanation

Solution

Dictionary order (Alphabetical order) mein letters is tarah honge:

A, H, M, S, T

Humein THAMS tak pahunchne ke liye har alphabet se shuru hone wale words ko count karna hoga:

Step 1: 'A' se shuru hone wale words

Agar pehla letter 'A' hai, toh bache hue 4 letters ko $4!$ tariko se arrange kiya ja sakta hai:

A _ _ _ _ \implies 4! = 24

Step 2: 'H' se shuru hone wale words

H _ _ _ _ \implies 4! = 24

Step 3: 'M' se shuru hone wale words

M _ _ _ _ \implies 4! = 24

Step 4: 'S' se shuru hone wale words

S _ _ _ _ \implies 4! = 24

Step 5: 'T' se shuru hone wale words

Ab hum 'T' par pahunch gaye hain. Ab second letter ko alphabetical order mein check karenge:

'TA' se shuru hone wale words:

$TA _ _ _ \implies 3! = 6$
'TH' se shuru hone wale words:

Yahan 'TH' match kar gaya hai. Ab third letter check karenge:
- 'THA' se shuru hone wale words:
  
  Yahan 'THA' bhi match kar gaya hai. Ab fourth letter check karenge:
  - 'THAM' se shuru hone wale words:
    
    Yahan 'THAM' bhi match kar gaya hai. Ab last letter check karenge:
    - Pehla word hoga: THAMS (Yeh wahi word hai jo humein chahiye).

Final Calculation

Serial Number (Rank) nikalne ke liye sabhi counts ko jod dete hain:

Rank = (\text{Words starting with A, H, M, S}) + (\text{Words starting with TA}) + 1 \ (\text{for THAMS})

Rank = (24 + 24 + 24 + 24) + 6 + 1

Rank = 96 + 6 + 1

Rank = 103

Serial number of the word THAMS is $103$ .

Choose the correct answer:

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