Permutation and Combination

Here is a simple problem:

How many sequences {a1, a2, a3, a4, a5} are there such that:

1. a1<=20, a2<=40, a3<=60, a4<=150, a5<=300
2. LCM(a1, a2, a3, a4, a5) = MAX(a1, a2, a3, a4, a5) = 210

MAX (a, b, c, d, e) is the largest of these 5 numbers.

Permutation and Combination

Here's a super interesting problem for you. Its been really hard to find such pretty problems.  This can be solved with JEE knowledge, so you should be able to do it :) .

We want to create a 'Divisible Sequence' of length H from a number N. In a Divisible Sequence, every term (except the starting number) is a divisor of the previous term. Examples of Divisible Sequences of length 3 starting with 10 are:
10, 10, 5
10, 2, 2
10, 10, 1
10, 1, 1

As you can see, there are many Divisible Sequences of length 3 starting with 10. Tell me the number of Divisible Sequences of length 10 starting with 264600.

N cakes are arranged in a line. Each cake is of a different flavor. You and your friend play a game to eat the cakes. When you eat a cake, your friend eats the (<=2) adjacent cakes.

For example, suppose there were 5 cakes initially named A,B,C,D and E. If you eat cake C, your friend eats  B and D, and the new configuration is A,E. Now, if you eat cake A, your friend eats cake E.

The question is simple: how many possible combinations of cakes can you finally eat? Note that the order of eating does not matter for the answer.

Hint: consider the Nth Fibonacci number, also called FN. Everyone knows that:

1. F1=1, F2=2 and Fn=Fn-1+Fn-1
2. FN is the number of ways to choose some objects from N objects (arranged in a line) such that no 2 are adjacent.
3. FN is the also number of ways to climb a flight of n stairs using one or 2 steps.