# Math Help - [SOLVED] Permutations and combinations question..

1. ## [SOLVED] Permutations and combinations question..

Each of the digits, 1, 1, 2, 3, 3, 4, 6 is written on a separate card. The seven cards are then laid out in a row to form a 7-digit number.

(a) how many distinct 7-digit numbers are there? [answered - might be related, so posting here]

7! / 2!x2! = 1260

(d) How many of these 7-digit numbers start and end with the same digit?

Do we need to consider distinct objects in this question? I don't get it. I have tried different ways but all failed.

2. Originally Posted by struck
Each of the digits, 1, 1, 2, 3, 3, 4, 6 is written on a separate card. The seven cards are then laid out in a row to form a 7-digit number.

(a) how many distinct 7-digit numbers are there? [answered - might be related, so posting here]

7! / 2!x2! = 1260

(d) How many of these 7-digit numbers start and end with the same digit?

Do we need to consider distinct objects in this question? I don't get it. I have tried different ways but all failed.

5!x2

4. Hello, struck!

Each of the digits, 1, 1, 2, 3, 3, 4, 6 is written on a separate card.
The seven cards are then laid out in a row to form a 7-digit number.

(a) How many distinct 7-digit numbers are there?
. . . $\frac{7!}{2!\,2!} \:=\:1260$ . . . . Right!

(d) How many of these 7-digit numbers start and end with the same digit?
You and mathaddict are both correct . . .

[1] Numbers that begin and end with 1: . $1\:\_\:\_\:\_\:\_\:\_\:1$
. . The other five digits (2,3,3,4,6} can be arranged in: ${5\choose2} \,=\,60$ ways.

[2] Numbers that begin and end with 3: . $3\:\_\:\_\:\_\:\_\:\_\:3$
. . The other five digits {1,1,2,4,6} can be arranged in: ${5\choose2} \,=\,60$ ways.

Therefore, there are: . $60 + 60 \:=\:{\color{blue}120}$ numbers that begin and end with the same digit.