1)Seven blank spaces numbered 1 to 7, are arranged in a row in ascending from left to right.Two As',three Bs'and two Cs' are to be filled in these 7 blank spaces such no two (similar)alphabet occupy consecutive positions.In how many ways can this be done if B occupy the first place?

 

2)Three distinct numbers are randomly selected from first 20 natural numbers .Find the probability that these numbers are in a geometric progression?

 

Hello raja987654321

 

Please post only one querry at once.

 

(i) As per your query there are seven blank spaces and B in the first position

then the query loks like as :

 

1	2	3	4	5	6	7
B

A

B

C

A

B

C

 

In this arrangement the letters in first batch as well as in second batch can be arranged as 3! manner so total arrangements are 3! * 3! = 36 arrangements.

 

Now as in the first set there are two cases where C will be at last place and number of arragements where C be at starting position in second sets are 2! = 2 so there are 2*2 = 4 possible arrangements where C must be come together. Same way when B at last place and A at last place in first set then there are again 2 arrangements possible where each letter at startig in the second set so totol 12 arrangements are possible where letters occupy consecutive positions.

 

Now total requisite arrangements are 36 - 12 = 24

 

Again in the first set there are 2! ways where B in the Starting position so we should avoid these cases too.

 

 

 

the answer for second question is 2/285

selecting any three numbers  can be done in 20 c 3 ways

 

i.e 1140

 

geometric progression can be formed only with 2, 3 or 4 as ratio

 

with 2 we have 5 series

with 3 we have 2 series and with 4 we have 1 series

 

total is eight hence ratio is 8/1140

i.e 2/285