Define an integer k to be odd if k 1 is even. Write up a formal proof of the following, using an indirect proof: Claim: For any natural numbers m and n, if m is odd and n is odd, then m n is even. (Notice--this is not hard to prove, but make sure that you are using a proof by contradiction. And use the official definitions of even and odd.)
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.
Prove the following:
For any natural number n, if n 2 + 6 n is an even number, then n is an even number.
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