Question

Seven people leave their bags outside temple and while returning after worshiping the deity, picked one bag each at random. In how many ways at least one and at most three of them get their correct bags?

• A. ${ }^{7} C _{3} \cdot 9+{ }^{7} C _{5} \cdot 44+{ }^{7} C _{1} \cdot 265$
• B. ${ }^{7} C _{6} \cdot 265+{ }^{7} C _{5} \cdot 9+{ }^{7} C _{7} \cdot 44$
• C. ${ }^{7} C _{5} \cdot 9+{ }^{7} C _{2} \cdot 44+{ }^{7} C _{1} \cdot 265$
• D. None of these

Answer 1

Answer: (A)

No. of ways getting one correct
$={ }^{7} C_{1} 6 !\left(1-\frac{1}{1 !}+\frac{1}{2 !}-\frac{1}{3 !}+\ldots+\frac{1}{6 !}\right) $
$={ }^{7} C_{1} \cdot 265$
No. of ways getting two correct
$={ }^{7} C _{2} \cdot 5 !\left(1-\frac{1}{1 !}+\frac{1}{2 !}-\frac{1}{3 !}+\ldots-\frac{1}{5 !}\right) $
$={ }^{7} C _{2} \cdot 44$
No. of ways getting three correct
$={ }^{7} C_{3} \cdot 4 !\left(1-\frac{1}{1 !}+\frac{1}{2 !}-\frac{1}{3 !}+\frac{1}{4 !}\right) $
$={ }^{7} C_{3} \cdot 9$
Required no. of ways
$={ }^{7} C_{3} \cdot 9+{ }^{7} C_{2} \cdot 44+{ }^{7} C_{1} \cdot 265$