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Rule of Sum:

If a first task can be perfromed in m ways while a second can be performed in n ways and the two tasks cannot be performed simultaneously then: performing either task can be accomplished in any one of m + n ways.

Rule of Product:

If a procedure can be broken down into first and secon stages and if there are m possible outcomes for the first stage and if, for each of these outcomes, there are n possible outcomes for the second stage, then the total procedure can be carried out in the designated order in mn ways.

Permutations

If there are n distinct objects, denoted a₁a₂…a_n and r is an integer with 1 ≤ r ≤ n, then by the rule of product, the number of permutations of size r for the n objects is:
P(n,r) = n ⋅ (n-1) ⋅ (n-2) ⋅ … ⋅ (n-r+1) = n!/(n-r)!

note: here no repetitions are allowed. If repetitions had been allowed, n^r using the rule of product
note: if using all then r=n and P(n,n) = n!

If there are n objects with n₁ of a first type, n₂ type … and n_r of an rth type where n₁ + n₂ + … + n_r = n then there are: n!/(n₁!n₂!…n_r!) linear arrangements of the given n objects.
ex: PEPPER

note: here objects of the same type are indistinguishable
note: for a circular arrangemnt (like position around a table) fix A the top point and solve by seating the rest.

n choose r
If we start with n distinct objects, each selection or combination of r of these objects with no reference to order corresponds to r! premutations of size r from a collection of size denoted C(n,r) where 0 ≤ r ≤ n satisfies r! ⋅ C(n,r) = P(n,r) and

C(n,r) = P(n,r)/r! = n!/(r!(n-r)! 0 ≤ r ≤ n

C(n,0) = 1 for all n ≥ 0
C(n,r) = C(n,n-r)