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Maths Question 8 – JEE-MAIN 2026

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The number of ways, of forming a queue of 4 boys and 3 girls such that all the girls are not together, is:

To find the number of ways where a specific condition (all girls are not together) is met, it's often easier to calculate the total number of ways and subtract the number of ways where the complementary condition (all girls are together) is met.

Step 1: Calculate total unrestricted arrangements✦ Active

There are 4 boys and 3 girls, making a total of 4+3=7 people. The total number of ways to arrange these 7 distinct people in a queue without any restrictions is given by 7!.

7!=7×6×5×4×3×2×1=5040
Step 2: Calculate arrangements where all girls are together○ Expand

To find the number of ways where all 3 girls are together, treat the 3 girls as a single unit. Now we are arranging 4 boys and this one unit of girls, which is a total of 4+1=5 units. These 5 units can be arranged in 5! ways. Additionally, the 3 girls within their unit can be arranged among themselves in 3! ways.

Number of ways=5!×3!=(5×4×3×2×1)×(3×2×1)=120×6=720
Step 3: Calculate arrangements where all girls are not together○ Expand

The number of ways where all the girls are not together is the total number of arrangements minus the number of arrangements where all the girls are together.

Number of ways=Total arrangements−Arrangements where girls are together=5040−720=4320
💡 Teacher's Secret Hint

This method of complementary counting is efficient for 'not together' conditions.

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