What is the probability of throwing the same number exactly 3 times?

[Dell]

we throw a dice 5 times,
what is the probability of throwing the same number exactly 3 times?

i tried using all kinds of techuniques but can't seem to get this,

if A is a given number and X is any other number

A* A* A* X* X
6* 1* 1* 5* 5 =150

150/6^5=0.0192901

but that doesn't take into account any other options, for example A*X*A*X*A, X*A*A*X*A ...

so then i thought since i can either roll a given number or not, and the possibility that i will is 1/6

X~B(5,1/6)

P(X=3)=5C3*(1/6)^3*(5/6)^2=0.03215

the correct answer is 0.19,- which makes me think my 1st attempt was correct and there are 10 options, but how do i find these options and also, why is the binomial incorrect??

 

[Mentallic]

If you take one possible sequence as you've done, giving AAAXX with A being the number of choice, what is the probability of achieving this sequence? How many other possibilities are there to mix up the order of the above sequence, such as AAXAX etc. ? And finally, how many possible numbers for A are there?

The same result can be achieved by the binomial distribution method.

 

[Dell]

thanks i got it, i found the probability of a cerain number 3 times, i need to multiply by 6

 

[Mentallic]

You got to love those silly mistakes