Binomial Theorem

The mean of a binomial distribution is x, and the variance is 5 .
We know that the mean, µ=np=x. This means n times p is x.
The formula for variance is σ2=np(1−p)=5. This means n times p times (1−p) is 5 .
From the mean formula, p=‌

x

n

.
Substitute p=‌

x

n

into the variance formula:
x(1−‌

x

n

)=5
If we open the bracket, we get: x−‌

x2

n

=5
Take ‌

x2

n

to the other side: ‌

x2

n

=x−5
Now, solve for n : n=‌

x2

x−5

To get a whole number for n,x−5 must divide x2 exactly. Let's check different x values:
If x=6:n=‌

36

1

=3636 is an integer.
If x=10:n=‌

100

5

=2020 is an integer.
If x=30:n=‌

900

25

=3636 is an integer.
So, the possible values for x are: 6,10,30.