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

There are 3 cases
a)1 and 2 are together
b) 1 and 2 have one digit in between
c) 1 and 2 have 2 digits in between

Case a
1 and 2 are together
so we have 6 digits now with 1 and 2 together=6!* 2!
Case b
1 and 2 have one digit in between
1 _ 2 this blank can be filled in 5C1 ways
5!*5C1*2!
Case c
1 and 2 have two digits in between
1 _ _2 these blanks can be filled in 5C2 ways with the digits interchanging 5C2*2!
total numbers =4!*5C2*2!*2!
So total ways= 6!* 2!+ 5!*5C1*2!+ 4!*5C2*2!*2!
=720*2+120*5*2+24*10*2*2
=1440+1200+960
=2640+960
=3600 ways

Answer 2

hey ash are you still there?

Answer 3

could you please explain case b) and case c)
i don't understand how you got 5!*5C1*2! and 5C2*2!

Answer 4

AAHHHHH I GET IT, I UNDERSTAND NOW hehehehe
THANK YOU for the very detailed explanation!!!!!!