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**1. Homework Statement**

Bowman shoots into a dartboard, with possible gain ranging from 0 to 10 points.

Probability that he shoots 30 points in 3 shots is 0.008.

Probability that he shoots < 8 in one shot is 0.4.

Probability that he shoots exactly 8 in one shot is 0.15.

What is the probability that he gains at least 28 points in 3 shots?

**3. The Attempt at a Solution**

My solution:

[tex]

P(X \ge 28) = P(X = 28) + P(X = 29) + P(X = 30)

[/tex]

We know

**P(X = 30)**so it's sufficient to count

**P(X = 28)**and

**P(X = 29)**.

__X = 28__This situation can occur either if:

(a) He shoots 10, 10 and 8 (in any order)

(b) He shoots 10, 9, 9 (in any order)

So I guess:

[tex]

(*)\ \ \ \ P(X = 28) = (P(X = 10).P(X =10).P(X = 8))+(P(X=10).P(X=9).P(X=9))

[/tex]

What I'm interested in is whether this is ok. I don't know if I should take into an account that (let's take for example the case

**(a)**) the bowman can shoot the points in any order, ie. 10, 10, 8 or 10, 8, 10 and so on.

Don't I have to multiply

**(*)**with 3! so that I cover all the orders in which the shooter can gain those points?

Thank you very much.