How many ways are there to give 5 apples and 7 bananas to 12 people assuming tha

Question

How many ways are there to give 5 apples and 7 bananas to 12 people assuming that each person gets a piece of fruit. How many ways can 7 oranges, 4 pears and 5 bananas be distributed to 16 people if each person gets one piece of fruit? How many ways are there of distributing 6 apples and 12 oranges to 20 people with no restriction on t lie number of apples or oranges a person can get. (d) You are asked to write down all integers from 1 to 10^n. n elementof N. How many times did you write down the digit 3.

Explanation / Answer

(A)

5 people from 12 to who apples will be given in 12C5 ways. This means there is only one way possible that remaining 7 will get bananas.

Hence required number of ways = 12C5 * 1 = 792

(B)

7 people from 16 people to who oranges will be given in 16C7 ways. And 4 people from remaining 9 people can be selected in 9C4 ways, these are the people who receive pears. This means there is only one to way to select remaining 5 people to give bananas.

Hence required number of ways = 16C7 * 9C4 * 1 = 1441440

(C)

here the equation would be

x1 + x2 + .... + x20 = 18

where x1, x2, ... x20 >= 0 but <=18

In order to find the total number of solutions to this equation, this can be done in (18 + 20 -1 ) C (20 - 1) = 37C19

(D)

Let f(n) be the number of digit 3 that will appear.

For n = 1, f(n) = 1 and for n=2, f(n) = 18

For n>2, we can have the below funtion

f(n) = f(n - 1)*10 + 10^(n-1)

Using this function, for n = 3, f(n) = 180 + 100 = 280

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