Answer:
(a) 10,626 different arrangements
(b) 95,367,431,640,625 different arrangements
(c) 2.5852017 × 10²² different arrangements
Step-by-step explanation:
(a) How many different arrangements are there if you only care about the number of books on the shelves (and not which book is where)?
We use the combination formula for this
C(n , r) = n + r - 1C r - 1
n = 20
r = 5
= 20 + 5 - 1 C 5-1
= 24C4
= 24!/4 ! × (24 - 4)
= 24!/4! × 20!
= 10,626 arrangements
(b) How many different arrangements are there if you care about which books are where, but the order of the books on the shelves doesn't matter?
Since the order of the books on the shelves does not matter,
The calculation is given as
5²⁰ = 95,367,431,640,625 arrangements
(c) How many different arrangements are there if the order on the shelves does matter?
Since order matters now
Step 1
We use the combination formula for this
C(n , r) = n + r - 1C r - 1
n = 20
r = 5
= 20 + 5 - 1 C 5-1
= 24C4
= 24!/4 ! × (24 - 4)
= 24!/4! × 20!
= 10,626
Step 2
We find the factorial of the number books
= 20!
= 2,432,902,008,176,640,000
Step 3
The different arrangements there are if the order on the shelves does matter is calculated
= 10,626 × 2,432,902,008,176,640,000
= 2.5852017 × 10²² different arrangements
