Question

A random draw is being designed for 210 participants. A single winner is to be chosen, and all the participants must have an equal probability of winning. If the winner is to be drawn using 10 balls numbered 0 through 9, how many balls need to be picked, regardless of order, so that each of the 210 participants can be assigned a unique set of numbers?

a) 10
b) 4
c) 5
d) 3

Answer 1

Option b - 4

Explanation:

Given : A random draw is being designed for 210 participants.

A single winner is to be chosen, and all the participants must have an equal probability of winning.

If the winner is to be drawn using 10 balls numbered 0 through 9.

To find : How many balls need to be picked, regardless of order, so that each of the 210 participants can be assigned a unique set of numbers?

Solution :

Let n be the number of balls drawn.

According to the question,

n balls are to be drawn out of the 10 balls such that we get total 210 choices irrespective of their order i.e.
`^(10)C_n=210`

Now we check fro given options,

a) The value of n=10

`^(10)C_(10)=(10!)/(10!* 0!)`

`^(10)C_(10)=1\\eq 210`

It is not correct.

b) The value of n=4

`^(10)C_(4)=(10!)/(4!* (10-4)!)`

`^(10)C_(4)=(10* 9* 8* 7* 6!)/(4* 3* 2* 6!)`

`^(10)C_(4)=210`

It is correct.

c) The value of n=5

`^(10)C_(5)=(10!)/(5!* (10-5)!)`

`^(10)C_(5)=(10* 9* 8* 7* 6* 5!)/(5*4* 3* 2* 5!)`

`^(10)C_(4)=252\\eq 210`

It is not correct.

d) The value of n=3

`^(10)C_(3)=(10!)/(3!* (10-3)!)`

`^(10)C_(3)=(10* 9* 8* 7!)/(3* 2* 7!)`

`^(10)C_(4)=120\\eq 210`

It is not correct.

Therefore, option b is correct.

Answer 2

Answer: The correct number of balls is (b) 4.

Step-by-step explanation: Given that a single winner is to be chosen in a random draw designed for 210 participants. Also, there is an equal probability of winning for each participant.

We are using 10 balls, numbered through 0 to 9. We are to find the number of balls which needs to be picked up, regardless of order, so that each of the 210 participants can be assigned a unique set of numbers.

Let 'r' represents the number of balls to be picked up.

Since we are choosing from 10 balls, so we must have

`^(10)C_r=210.`

The value of 'r' can be any one of 0, 1, 2, . . , 10.

Now,

if r = 1, then

`^(10)C_1=(10!)/(1!(10-1)!)=(10!)/(1!9!)=(10* 9!)/(1* 9!)=10<210.`

If r = 2, then

`^(10)C_2=(10!)/(2!(10-2)!)=(10!)/(2!8!)=(10* 9* 8!)/(2* 1* 8!)=45<210.`

If r = 3, then

`^(10)C_3=(10!)/(3!(10-3)!)=(10!)/(3!7!)=(10* 9* 8* 7!)/(3* 2* 1* 7!)=120<210.`

If r = 4, then

`^(10)C_4=(10!)/(4!(10-4)!)=(10!)/(4!6!)=(10* 9* 8** 7* 6!)/(4* 3* 2* 1* 6!)=210.`

Therefore, we need to pick 4 balls so that each participant can be assigned a unique set of numbers.

Thus, (b) is the correct option.