Twenty documents must be moved from a computers desktop into

Twenty documents must be moved from a computer’s desktop into folders A or B. The order in which these documents are moved is irrelevant, and no document must be in both folders.

a) Find the number of ways this can be done. b) Find the number of ways this can be done if no folder can remain empty.

b) Find the number of ways this can be done if no folder can remain empty.

Solution

20 documents have to be moved into folders A and B

Where the order of moving is irrelevant

and no documents must be in both folders

a)

TO FIND: no. of ways this can be dome

CASE 1 ---

All 20 documents in folder A and 0 documents in B

this can be done in 20! * 0! ways

CASE 2 ---

19 documents in A and 1 document in B

this can be done in 19! * 1! ways

CASE 3 ---

18 documents in A and 2 documents in B

This can be done in 18! * 2! ways

Likewise we continue till we finally get

CASE 19 ---

2 documents in A and 18 documents in B

this can be done in 2! * 18! ways

CASE 20 ---

1 document in A and 19 documents in B

this can be done in 1! * 19! ways

CASE 21 ---

0 document in A and 20 documents in B

this can be done in 0! * 20! ways

Now we add up all these schenarios to get the total number of ways

= (20! 0!) + (19! 1!) + (18! 2!) +(17! 3!) + (16! 4!) + (15! 5!) + (14! 6!) + (13! 7!) + (12! 8!) + (11! 9!) + (10! 10!) + (9! 11!) + (8! 12!) + (7! 13!) + (6! 14!) + (5! 15!) + (4! 16!) + (3! 17!) + (2! 18!) + (1! 19!) + (0! 20!)

= 2 [ (20! 0!) + (19! 1!) + (18! 2!) +(17! 3!) + (16! 4!) + (15! 5!) + (14! 6!) + (13! 7!) + (12! 8!) + (11! 9!) ] + (10! 10!)

= 2 [ 20! + 19! + (18! 2!) +(17! 3!) + (16! 4!) + (15! 5!) + (14! 6!) + (13! 7!) + (12! 8!) + (11! 9!) ] + (10! 10!)

taking common factorial from 2 terms combined

= 2 [ 19! (20 + 1) + 17! 2! (18 + 3) + 15! 4! (16 + 5) + 13! 6! (14 + 7) + 11! 8! (12 + 9) ] + (10! 10!)

= 2 [ (19! 21) + (17! 2! 21) + (15! 4! 21) + (13! 6! 21) + (11! 8! 21)] + (10! 10!)

=2 * 21 [ 19! + 17! 2! + 15! 4! + 13! 6! + 11! 8!] + (10! 10!)

= 42 [ 19! + 17! 2! + 15! 4! + 13! 6! + 11! 8!] + (10! 10!)

b) TO FIND: no. of ways this can be done if no folder can remain empty

CASE 1 ---

19 documents in A and 1 document in B

this can be done in 19! * 1! ways

CASE 2 ---

18 documents in A and 2 documents in B

This can be done in 18! * 2! ways

Likewise we continue till we finally get

CASE 18 ---

2 documents in A and 18 documents in B

this can be done in 2! * 18! ways

CASE 19 ---

1 document in A and 19 documents in B

this can be done in 1! * 19! ways

Now we add up all these schenarios to get the total number of ways

= (19! 1!) + (18! 2!) +(17! 3!) + (16! 4!) + (15! 5!) + (14! 6!) + (13! 7!) + (12! 8!) + (11! 9!) + (10! 10!) + (9! 11!) + (8! 12!) + (7! 13!) + (6! 14!) + (5! 15!) + (4! 16!) + (3! 17!) + (2! 18!) + (1! 19!)

= 2 [ (19! 1!) + (18! 2!) +(17! 3!) + (16! 4!) + (15! 5!) + (14! 6!) + (13! 7!) + (12! 8!) + (11! 9!) ] + (10! 10!)

= 2 [ 19! + (18! 2!) +(17! 3!) + (16! 4!) + (15! 5!) + (14! 6!) + (13! 7!) + (12! 8!) + (11! 9!) ] + (10! 10!)

= 2 [ 19! + 17! 2! (18 + 3) + 15! 4! (16 + 5) + 13! 6! (14 + 7) + 11! 8! (12 + 9) ] + (10! 10!)

= 2 [ 19! + (17! 2! 21) + (15! 4! 21) + (13! 6! 21) + (11! 8! 21)] + (10! 10!)

= 2 * 21 [ 19!/21 + 17! 2! + 15! 4! + 13! 6! + 11! 8!] + (10! 10!)

= 42 [ 19!/21 + 17! 2! + 15! 4! + 13! 6! + 11! 8!] + (10! 10!)