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Artist 52 [7]
3 years ago
10

Twenty different books are to be put on five book shelves, each of which holds at least twenty books.

Mathematics
1 answer:
olya-2409 [2.1K]3 years ago
8 0

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

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If the unknown value is x, then x = \frac{14}{4} × 6 = 21.

Explanation:

  • Since it is a double number line with a ratio, all the ratios of one number line to the other will remain equal for all the values along both the number lines. Assume the unknown value to be x.
  • The ratio of the number on the first number line to the corresponding number on the second number line is calculated to be;                                  \frac{7}{2} = \frac{14}{4} = \frac{x}{6} = \frac{28}{8} = \frac{35}{10} = 3.5. So \frac{x}{6} = 3.5, x = 3.5 × 6 = 21.
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