Question

how many numbers of 7 digits can be formed from the digits 1,2,3,4,5,6,7 if there are not more than 2 digits between 1 and 2?

Answer 1

I think 1800

Answer 2

Assuming digits cannot be repeated,
look at case where 1 and 2 are next to each other, case where single digit is in between and case where 2 digits are in between.

case 1:
For the "12" grouping there are 6! ways of arranging them
For the "21" grouping there are also 6! ways of arranging them
= 2*6!

case 2:
For the "1?2" grouping there are 5 possible digits to go in between and 5! ways of arranging all the digits...5*5!
For the "2?1" grouping there are 5 possible digits to go in between and 5! ways of arranging all the digits...5*5!
= 10*5!

case 3:
For the "1? ? 2" grouping there are 5 choose 2 possible groups to go in between and 4! ways of arranging all the digits...10*4!
For the "2? ? 1" grouping there are 5 choose 2 possible groups to go in between and 4! ways of arranging all the digits...10*4!
= 20*4!

Total numbers is = 2*6! + 10*5! + 20*4!

Answer 3

thank you ^^

Answer 4

@dumbcow: "For the "1? ? 2" grouping there are *5 choose 2* possible groups to go in between"

why \( \binom{5}{2} \) and not \( ^5P_2 \)?

Answer 5

I am getting \(40×3!\), consistent to \( 4!× ^5P_2×2! \) for the last case.

Answer 6

LOL its ok foolformath, i have figured out how to do this question
but nevertheless your help is greatly appreciated!

Answer 7

Wait, I want to know if I am wrong or not.

Answer 8

its actually 4!× ^5P_2×2! x2!

Answer 9

From where do you get the additional \( 2! \) ?

Answer 10

ok that came out wierd
but basically is what you had x 2!

Answer 11

because you can interchange the 1 and 2

Answer 12

but then how you explain the other 2! ?

Answer 13

oh because when you have 5c2, these two no.s can swap since there are two no.s.
thats why in the case where you only had 1 no. between 1 and 2 you DID NOT have the ADDITIONAL 2!

Answer 14

Exactly, dumbcow's answer needs another \(2! \) to be correct.

Answer 15

GOOD detective skills, i didn't even notice that hehehe!

Answer 16

Lol, thanks :)

Answer 17

Ahh yes i missed that...thanks for correcting me

Answer 18

anytime ^.^