Let m and n be two relatively prime positive naturals, and consider what naturals can be expressed as linear combinations am + bn where a and b are naturals, not just integers. (a) Show that if m=2 and n=3,any natural except 0 and 1 can be so expressed (b) Determine which naturals can be expressed if m = 3 and n = 5. (c) Argue that for any m and n, there are only a finite number of naturals that cannot be expressed in this wa
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.
Let m and n be two relatively prime positive naturals, and consider what naturals can be expressed as linear combinations am + bn where a and b are naturals, not just integers.
(a) Show that if m=2 and n=3,any natural except 0 and 1 can be so expressed
(b) Determine which naturals can be expressed if m = 3 and n = 5.
(c) Argue that for any m and n, there are only a finite number of naturals that cannot be expressed in this way.
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 3 images
