Question 8 How many different ways can 100 pennies be distributed among 10 peopl

Question

Question 8 How many different ways can 100 pennies be distributed among 10 people so that: a. everyone gets 0, or more, pennies? b. everyone gets 5, or more, pennies? Question 9 There are 30 identical diamonds buried on an island. You have a treasure map of the island that tells you that the diamonds are distributed among 9 different locations. a. What's the maximum number of different locations that could have b. How many different ways could the diamonds be distributed among c. Suppose you dig in the first three locations, and discover 0, 4, and 0 diamonds? the locations (assuming each location has 0 or more diamonds)? 2 diamonds. How many different ways can the remaining diamonds be distributed among the remaining locations?

Explanation / Answer

#8.
There are 10 different variables in the left hand side of the equation and 100 on the right hand side

a)
x1 + x2 + . . . . + x10 = 100
In order to find the solution to this (100 + 10 - 1)C(10-1)

Required number of ways = 109C9 = 4,263,421,511,271

b)
here minimum value of every variable is 5, this mean there are only 100 - 5*10 = 50 pennies to be distributed

In order to find the solution to this (50 + 10 - 1)C(10-1)

Required number of ways = 69C9 = 56672074888

#9.
Here total 9 variables and summing it to 30
i.e. x1 + x2 + .... .. + x9 = 30

a)
If one location has all the 30 diamonds buried. There would be 8 other locations which will have 0 diamonds

b)
In order to find the solution to this (30 + 9 - 1)C(9-1)

Required number of ways = 38C8 = 48903492

c)
x1 = 0, x2 = 4 and x3 = 2
Now, remaining 30 - 6 = 24 diamonds are present in other 6 locations.

In order to find the solution to this (24 + 6 - 1)C(6-1)

Required number of ways = 29C5 = 118755

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