Question

In how many ways, 6 letters can be placed in 6 envelopes such that at least 4 letters go into their corresponding envelopes?

Answer 1

Answer: The answer is 22.

Step-by-step explanation: Given that there are 6 letters that can be placed in 6 envelopes. We are to find the number of ways in which at least 4 letters go into their corresponding envelopes.

According to the given information, we have

number of ways in which exactly 4 letters go into their corresponding envelopes is

$n_1=\dfrac{6!}{4!(6-4)!}=\dfrac{6\times 5\times 4!}{4!\times 2\times 1}=15,$

number of ways in which exactly 5 envelopes go into their corresponding envelopes is

$n_2=\dfrac{6!}{5!(6-5)!}=\dfrac{6\times 5!}{5!\times 1}=6,$

and the number of ways in which all the envelopes go into their corresponding envelopes is

$n_3=\dfrac{6!}{6!(6-6)!}=\dfrac{6!}{6!\times 1}=1.$

Therefore, the total number of ways is

$n=n_1+n_2+n_3=15+6+1=22.$

Thus, the answer is 22.