Respuesta :
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
[tex]^{10}C_r=210.[/tex]
The value of 'r' can be any one of 0, 1, 2, . . , 10.
Now,
if r = 1, then
[tex]^{10}C_1=\dfrac{10!}{1!(10-1)!}=\dfrac{10!}{1!9!}=\dfrac{10\times 9!}{1\times 9!}=10<210.[/tex]
If r = 2, then
[tex]^{10}C_2=\dfrac{10!}{2!(10-2)!}=\dfrac{10!}{2!8!}=\dfrac{10\times 9\times 8!}{2\times 1\times 8!}=45<210.[/tex]
If r = 3, then
[tex]^{10}C_3=\dfrac{10!}{3!(10-3)!}=\dfrac{10!}{3!7!}=\dfrac{10\times 9\times 8\times 7!}{3\times 2\times 1\times 7!}=120<210.[/tex]
If r = 4, then
[tex]^{10}C_4=\dfrac{10!}{4!(10-4)!}=\dfrac{10!}{4!6!}=\dfrac{10\times 9\times 8\times\times 7\times 6!}{4\times 3\times 2\times 1\times 6!}=210.[/tex]
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.
Answer:
Option b - 4
Step-by-step 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. [tex]^{10}C_n=210[/tex]
Now we check fro given options,
a) The value of n=10
[tex]^{10}C_{10}=\frac{10!}{10!\times 0!}[/tex]
[tex]^{10}C_{10}=1\neq 210[/tex]
It is not correct.
b) The value of n=4
[tex]^{10}C_{4}=\frac{10!}{4!\times (10-4)!}[/tex]
[tex]^{10}C_{4}=\frac{10\times 9\times 8\times 7\times 6!}{4\times 3\times 2\times 6!}[/tex]
[tex]^{10}C_{4}=210[/tex]
It is correct.
c) The value of n=5
[tex]^{10}C_{5}=\frac{10!}{5!\times (10-5)!}[/tex]
[tex]^{10}C_{5}=\frac{10\times 9\times 8\times 7\times 6\times 5!}{5\times4\times 3\times 2\times 5!}[/tex]
[tex]^{10}C_{4}=252\neq 210[/tex]
It is not correct.
d) The value of n=3
[tex]^{10}C_{3}=\frac{10!}{3!\times (10-3)!}[/tex]
[tex]^{10}C_{3}=\frac{10\times 9\times 8\times 7!}{3\times 2\times 7!}[/tex]
[tex]^{10}C_{4}=120\neq 210[/tex]
It is not correct.
Therefore, option b is correct.
