Bhargava Vadlamani

The first n natural numbers, 1 to n, have to be arranged in a row from left to right. The n numbers are arranged such that there are an odd number of numbers between any two even numbers as well as between any two odd numbers. If the number of ways in which this can be done is 72, then find the value of n?

Working together, A and B can do a job in 6 days. B and C can do the same job in 10 days, while C and A can do it in 7.5 days. How long will it take if all A, B and C work together to complete the job?

If A finishes the work in a days then in one day A does 1/a part of the work. Similarly, B does 1/b and C does 1/c part. IF W is the total work. Then below are per day works W/a + W/b = W/6 W/b + W/c = W/10 W/c + W/a = W/7.5 Adding above equations we get 2(1/a+1/b+1/c) = 1/6+1/10+1/7.5 --- > 1/a + 1/b + 1/b = 1/5 all three together do 1/5th of work in a day. So, they take 5 days to finish the work.

The speed of a motor boat itself is 20 km/h and the rate of flow of the river is 4 km/h. Moving with the stream the boat went 120 km. What distance will the boat cover during the same time going against the stream? 

 I travel the first part of my journey at 40 kmph and the second part at 60 kmph and cover the total distance of 240 km to my destination in 5 hours. How long did the first part of my journey last? 

The first n natural numbers, 1 to n, have to be arranged in a row from left to right. The n numbers are arranged such that there are an odd number of numbers between any two even numbers as well as between any two odd numbers. If the number of ways in which this can be done is 72, then find the value of n?

n=17. This is quite tricky. 1 to n natural numbers can be even number of numbers or odd number of numbers. ex - 1 2 3 4 5 or 1 2 3 4 5 6 so here n could be 5 or 6. We don't know hence we consider both the cases. n ---> even . Then there will be n/2 even and n/2 odd numbers in list of n numbers. Now we need observe that when we try to arrange odd number of numbers in between two even or odd numbers we end up having consecutive even and odd number always. So, number of ways will be n/2! * n/2! .Now lets try to equate it to 72. We get nothing as n/2! * n/2! is a perfect square and 72 is not. n -- > odd. Then we will have (n+1)/2 odd numbers and (n-1)/2 even numbers in the list of n numbers. we apply the same logic here too:- (n-1)/2 ! * (n+1)/2 ! = 72 (n^2 - 1)/4 = 72 ---- > n = 17. Hence there are 17 numbers 1 2 3 .... 17. Let me know if you need more explanation.