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In the editorial group’s photograph of a school all the 5 teachers are to be seated in the front row. Four girls are to be in the second row and six boys in the third row. If the principal has a fixed seat in the first row, then how many arrangements are possible? 
  • a)
     237144 
  • b)
    251820 
  • c)
    502340 
  • d)
    72000 
  • e)
    2073600
Correct answer is option 'E'. Can you explain this answer?
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Step 1: Arrangements in the first row
  • There are 5 teachers to be seated in the first row.
  • The principal has a fixed seat, so only the other 4 teachers can be arranged.
  • Number of ways to arrange the 4 teachers in the remaining 4 seats = 4! = 24.

Step 2: Arrangements in the second row
  • There are 4 girls to be seated.
  • Number of ways to arrange 4 girls in 4 seats = 4! = 24.

Step 3: Arrangements in the third row
  • There are 6 boys to be seated.
  • Number of ways to arrange 6 boys in 6 seats = 6! = 720.

Step 4: Total number of arrangements
Total arrangements = (arrangements in first row) × (arrangements in second row) × (arrangements in third row)
= 4! × 4! × 6!
= 24 × 24 × 720
= 24 × (24 × 720)
First calculate 24 × 720 = 17280
Then multiply by 24: 17280 × 24 = 414,720
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