How many ways are there to distribute 7 coupons into 4 envelopesif a.the coupons
ID: 2964294 • Letter: H
Question
How many ways are there to distribute 7 coupons into 4 envelopesif
a.the coupons are identical and the envelopes are all different and can contain any number of coupns?
b.the coupons are identical and the envelopes are all different but all must be nonempty?
c.the coupons are all different and the envelopes are all different but all must be nonempty?
d.the coupons are all differnet and envelopes are all identical but all must be nonempty?
e.the coupons are all different and the envelopes are all identical and can contain any number of coupons?
Explanation / Answer
Coupons =7 enevelopes =4
a) the coupons are identical and the envelopes are all different and can contain any number of coupns?
It is similar as X1+X2+X3+X4 = 7
so no ways = 7+4-1 C 4 -1 = 10 C 3 = 120 .
b) the coupons are identical and the envelopes are all different but all must be nonempty?
first keep one coupon to each enevelop then keep remaining 4 coupons as follow
It is similar as X1+X2+X3+X4 = 4
so no ways = 4+4-1 C 4 -1 = 7 C 3 = 35 .
c. the coupons are all different and the envelopes are all different but all must be
nonempty?
it is similar as no of onto function from A->B
coupons (7) --> enevelop (4)
U can calculate the no of onto function using following formula
A--->B onto function |A|=m and |B|=n
n! X S(m,n) where S(m,n) is stirling no of second kind. pls check following link to know more about stirling no
http://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind
http://austinmohr.com/home/?page_id=431
so no ways is 4! X S(7,4) = 24 X 350 = 8400.
d.the coupons are all differnet and envelopes are all identical but all must be nonempty?
it is similar as stirling no so
no of ways = S(7,4) = 350.
e.the coupons are all different and the envelopes are all identical and can contain any number of coupons?
no enevelop is empty
no of ways = S(7,4)
only one enevelop is empty then
no ways = S(7,3)
exactly two enevelops are empty
no ways= S(7,2)
exactly three enevelops are empty
no ways= S(7,1)
all are empty
no of ways =1
so
Total no ways = S(7,4)+S(7,3)+S(7,2)+S(7,1) +1 = 350 + 301 + 63 + 1 + 1 = 716
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