00:01
In subpart a of this problem, we are asked to find out the number of 12 letter words formed by 7 bs and 5 a's such that all of the a's are separated.
00:21
Which means that in between two consecutive a's, there must be a b.
00:27
So let us write down the bs in such a way.
00:31
So we have space b space b space b space b space b space b space b and a space so let us check the number of b that we have written so this is the first b second b third b fourth b fifth b sixth b and seventh b and and we have only 7 b's.
01:01
So now let us look at how many positions for a are available.
01:06
So we have the first position, second position, third position, fourth position, fifth position, sixth, seventh and eighth.
01:14
So here we have eight positions available for a but we have only five a's available.
01:23
So therefore the number of ways that we can place the a's is given by 8c5.
01:29
So let us calculate this.
01:31
We have 8 factorial divided by 5 factorial times 3 factorial.
01:37
8 factorial can be written as 8 times 7 times 6 times 5 factorial.
01:43
The whole divided by 5 factorial times 3 times 2 times 1.
01:48
So here 5 factorial and 5 factorial get cancelled.
01:51
3 times 2 is 6 so it get cancelled with 6.
01:55
So we are left with 8 times 7 which is 56.
02:03
So therefore there are 56 different words that can be formed by making use of 7 bs and 5 a's such that all the a's are separated.
02:14
So this is the required answer for subpart a.
02:19
In subpart b we are asked to find out how many 5 digit numbers can be formed where no 2 consulate.
02:30
Consecutive numbers are the same...