Question:
(a) How many three-digit numbers can be
formed from the digits 0, 1, 2, 3, 4, 5, and 6 if
each digit can be used only once?
(b) How many of these are odd numbers?
(c) How many are greater than 330?
My Solution:
a)(7)(6)(5).
Don't know the answers of other 2.
Correct me if I am wrong. Thanks!!!
Best Answer
There are two possible answers to this depending of if the number can start with a 0 or not.
If it can then the number is $7 \times 6 \times 5 = 210$
However we don't usually write numbers with leading zeros. So assuming we don't allow a leading zero the answer is $6 \times 6 \times 5 = 180$
How many of these are odd?
Lets consider picking the numbers in a special order
First we need to pick a 1, 3 or 5 so in our first draw for the final digit so we have a choice of 3. Next we pick our first digit which can be any thing apart from 0 or the number we have just picked making a choice of 5 and for our final draw the middle digit we can pick any of the 5 remaining digits making $3 \times 5 \times 5 = 75$
Finally For the third one greater than 330 we have two ways to achieve this either draw a 4, 5 or 6 for the first digit then we don't care about the others making $3 \times 6 \times 5 = 90$
Alternatively we must draw a 3 for our first digit then 4, 5 or 6 for our second and finally any digit making $1 \times 3 \times 5 = 15$
Now since both of these groups have no overlap we can simply add them together to get $90 + 15 = 105$.