Replying to a PM

Use the Venn diagram to solve this question quickly (refer to the figure below).

Let, number of students who received no gift be \(x\)
Total students = Chocolates + Pens + Wine - (both Chocolate and Pen) - (both Chocolate and Wine) - (both Wine and Pen) + All three + \(x\)

Since Pens are placed evenly starting from the 2nd locker, \(\frac{400}{2}\) students received Pens i.e. \(200\)

Since Chocolates are placed in every 3rd locker starting from the 3rd locker only, \(\frac{400}{3}\) students received Chocolates i.e. \(133\)

Since Wine bottles are placed in every 5th locker starting from the 5th locker only, \(\frac{400}{5}\) students received Wine bottles i.e. \(80\)

So, Total students = Chocolates + Pens + Wine - (both Chocolate and Pen) - (both Chocolate and Wine) - (both Wine and Pen) + All three + \(x\)
400 = 133 + 200 + 80 - (both Chocolate and Pen) - (both Chocolate and Wine) - (both Wine and Pen) + All three + \(x\)
400 = 413 - (both Chocolate and Pen) - (both Chocolate and Wine) - (both Wine and Pen) + All three + \(x\)

Now, let us find out the number of students who received both items

Chocolate (every 3rd) and Pen (every 2nd)
LCM of 3 and 2 = 6
So, \(\frac{400}{6} = 66\) students received both Chocolate and Pen

Chocolate (every 3rd) and Wine bottle (every 5th)
LCM of 3 and 5 = 15
So, \(\frac{400}{15} = 26\) students received both Chocolate and Wine bottle

Pen (every 2nd) and Wine bottle (every 5th)
LCM of 2 and 5 = 10
So, \(\frac{400}{10} = 40\) students received both Pen and Wine bottle

What about the students who received all three gifts?
LCM of 2, 3 and 5 = 30
So, \(\frac{400}{30} = 13\) students received all three gifts

\(400 = 413 - (66) - (26) - (40) + 13 + x\)
\(x = 400 - 413 + 132 - 13\)
\(x = 532 - 426\)
\(x = 106\)

Hence, option C

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