Permutations And Combinations Question 313

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?

Options:

A) $ ^{7}C_3.9{{+}^{7}}C_5.44{{+}^{7}}C_1.265 $

B) $ ^{7}C_6.265{{+}^{7}}C_5.9{{+}^{7}}C_7.44 $

C) $ ^{7}C_5.9{{+}^{7}}C_2.44{{+}^{7}}C_1.265 $

D) None of these

Show Answer

Answer:

Correct Answer: A

Solution:

  • [a] No. of ways getting one correct $ {{=}^{7}}C_66!( 1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+…+\frac{1}{6!} ){{=}^{7}}C_1.265 $ No. of ways getting two correct $ {{=}^{7}}C_2.5!( 1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+…-\frac{1}{5!} ){{=}^{7}}C_2.44 $ No. of ways getting three correct $ {{=}^{7}}C_3.4!( 1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!} ){{=}^{7}}C_3.9 $ Required no. of ways $ {{=}^{7}}C_3.9{{+}^{7}}C_2.44{{+}^{7}}C_1.265. $