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@visheshsahni

No one is perfect here yaar and if you find a mistake in a solution and know the right answer, least thing to do is to share the solution. That's how a forum (should) function right ? :)
Corrected the question.

Q30) A teacher asks one of her students to divide a 30-digit number by 11. The number consists of six consecutive 1’s, then six consecutive 2’s, and likewise six 3’s, six 4’s and six 7’s in that order from left to right. The student inserts a three-digit number between the last 4 and the first 7 by mistake and finds the resulting number to be divisible by 11. Find the number of possible values of the three-digit number.

let A,B,C,D
let A got a rs
B got a * b rupees
C got a * b * c rupees
D got a * b * c * d rupees
a(1+b+bc+bcd)=70
a * (1+b+bc+bcd) = 2 * 35
this will be minimum when all values are as close as possible
1+b+bc+bcd=35
b(1+c+cd)=34
so b=2
1+c+cd=17
so c(1+d)=16
so c=4 and d=3
so a=2,b=2,c=4,d=3
so 2,4,16,48 these are values

Q29) In a group of four boys, each of them has a certain number of Re. 1 coins with him. Every boy has an amount that is an integral multiple of the amount possessed by every other boy who has an amount less than him. If the total amount with the four boys put together is Rs. 70, and no two boys have the same amount, what is the minimum possible amount (in Rs.) with the boy who has the maximum number of Rs. 1 coins with him?

Q28) How many arrangements can be made in a round table of 8 chairs if 2 of them like to sit opposite to each other ?

1step = 1 way =2^0
2 step = (1,1) , (2,0) = 2^1 way
3 step = (1,1,1) , (2,1) ,(1,2) ,(3)=2^2=4 way
4 step = 2^3= 8 way
5 step = 2^4 = 16 way
6 step = 2^5 = 32 way
now step 7th step will be sum of previous 6 steps= 63
8th step = sum of previous 6 step = 125

Q27) In how many ways can you climb up 8 steps if the minimum and maximum steps you can take at a time are 1 and 6 respectively?

t + t/12 = 9 (time taken in the return journey is 1/12th the time taken for the onward journey)
t = 108/13 = 8.30 hours
t + t/6 = 9.68 hours?

Q26) A teacher asks one of her students to divide a 30-digit number by 11. The number consists of six consecutive 1’s, then six consecutive 2’s, and likewise six 3’s, six 4’s and six 7’s in that order from left to right. The student inserts a three-digit number between the last 4 and the first 7 by mistake and finds the resulting number to be divisible by 11. Find the number of possible values of the three-digit number

Q25) Let S be the set of the first six natural numbers. If five numbers are picked randomly from S, what is the probability that the sum of the five numbers is divisible by 3?