How many three digit number can be formed?

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!!!

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 $90 + 15 = 105$.

Any of the 6 nonzero digits can be chosen for the hundreds position, and of the remaining 6 digits for the tens position, leaving 5 digits for the units position. So, there are (6)(5)(5) = 150 three digit numbers.

• It should be $6\times 6\times 5$ as you and other post suggested long ago. – user99914 Oct 3 '15 at 9:53
• I 110% agree with you, and till now I don't understand why the solution manual wrote it like this (the answer I wrote is copied from the solution manual !!!!!! ) .... stupid solution manual -_- :D – Ammar Rashed Oct 3 '15 at 10:04
• However that can be true if ' 0 ' is to be used :D – Ammar Rashed Oct 3 '15 at 10:05