6. Let s, be the number of different symmetric trains of length n which are built out of cars which are either 1 unit long or 2 units long-where, a symmetric train is one that is the same running backwards and forwards. Thus 212 is a symmetric 5-train but 122 is not. For example, s5 = 2 (table at the right). Find a for- mula for s, in terms of the Fibonacci numbers. Can you find a general argument for your formula? The 2 symmetric trains of length 5 1-1-1-1-1 2-1-2 %3D
Permutations and Combinations
If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.
Counting Theory
The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.
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