Question 1143267
a) How many three digit numbers can be formed from the digits 2,3,4,5,6,7 and 8? No digit repeats in the same number.
<pre>
There are 7 digits.

Choose the 1st digit any of the 7 digits.  That's 7 ways.
Having chosen 1 digit, and since no digits can be repeated,
there remain only 6 digits to choose from.

Choose the 2nd digit any of the remaining 6 digits.  That's 6 ways.
Having chosen 2 digits, and since no digits can be repeated,
there remain only 5 digits to choose from. 

Choose the 3rd digit any of the remaining 5 digits. That's 5 ways.

That's 7×6×5 = 7P3 = 210
</pre>
b) What is the probability of an even number?
<pre>
We first answer the question: "How many even three-digit numbers can
we have.  We choose the most restrictive thing first, which is the
3rd digit, for it must be even.

Choose the 3rd digit any of the 4 even digits 2,4,6,8.  That's 4 ways.
Having chosen 1 digit, and since no digits can be repeated,
there remain only 6 digits to choose from.

Choose the 1st digit any of the remaining 6 digits.  That's 6 ways.
Having chosen 2 digits, and since no digits can be repeated,
there remain only 5 digits to choose from.

Choose the 2nd digit any of the remaining 5 digits.  That's 5 ways.

So that's 4×6×5 = 120 ways

So the probability of an even number is 120 out of 720 or
120/720 which reduces to 1/6.
</pre>
c) What is the probability of numbers greater than 550 but less than 770?
<pre>
The first digit is either 5, 6 or 7, but we must take each of those
as a separate case because with each, the number of choices for the
2nd digit changes.

Case 1.  the first digit is 5.

Choose the first digit as 5. That's 1 way.
Choose the 2nd digit any of the 3 digits 6,7,8.  That's 3 ways.
Having chosen 2 digits, and since no digits can be repeated,
there remain only 5 digits to choose from for the 3rd digit.

That's 1×3×5 = 15 ways for case 1.

Case 2.  the first digit is 6.

Choose the first digit as 6.  That's 1 way.
Choose the 2nd digit as any of the 6 digits 2,3,4,5,7,8.  That's 6 ways.
Having chosen 2 digits, and since no digits can be repeated,
there remain only 5 digits to choose from for the 3rd digit.

That's 1×6×5 = 30 ways for case 2. 

Case 3.  the first digit is 7.

Choose the first digit as 7.  That's 1 way
Choose the 2nd digit any of the 5 digits 2,3,4,5,6.  That's 5 ways.
Having chosen 2 digits, and since no digits can be repeated,
there remain only 5 digits to choose from. That's 5 ways.

That's 1×5×5 = 25 ways for case 3.

So for all three cases, that's a grand total of 15+30+25 = 70 ways

So the probability of a number bettween 550 and 770 is 
70 ways out of 210 or 70/210 which reduces to 1/3.

Edwin</pre>