Combinatorial proof: Prove the following i*["n choose i"] = n*["n-1 choose i-1"].

Show the following i*["n choose i"] = n*["n-1 choose i-1"]. (See attached file for the more mathematical version.)

This problem looks at probabilities in terms of one minus the probability of the compliment in order to solve the problem: if a die is rolled 4 times, what is the probability that a 3 comes up at least once

If a die is rolled 4 times, what is the probability that a 3 comes up at least once?

Permutations and combinations

A set of 10 flags, 5 red, 3 blue and 2 yellow are to be arranged in a line along a balcony. If flags of the same colour are INDISTINGUISHABLE, find the number of arrangements in which, 1) The three blue flags are together 2) The yellow flags are not together 3) The red flags occupy alternate positions in the line 4) If the ...continues

Combinations : Seating Arrangements at a Dinner Table

In how many ways can 6 couples be seated at a circular table if each couple is not to be separated? How many ways can 5 Manchester United and 8 Chlesea players be seated at a circular dinner table if no two Manchester United players can sit together?

Permutations and combinations: Arrangements of letters in the word "PARALLELOGRAM".

Find the number of different selections of three letters which can be made from the letters of the word PARALLELOGRAM. How many of these contain the letter P?

Find the length difference of two moving objects.

L1=Lo*sqrt(1-(v/c)^2) where Lo is length of an obiect at rest, c is velosity of light, v is velocity of the object. If L2=Lo*(c^2-v^2)/c^2, what is L1-L2?

Combination

Q: How many four-digits numbers can be formed under the following conditions? (a) Leading digits cannot be zero. (b) Leading digits cannot be zero and no repetition of digits is allowed. (c) Leading digits cannot be zero and the number must be a multiple of 5.

Palindromes

What are the palindromes between 100 & 200? Find the range, mode, and median.

Skolem Sequences

Please see the attached file for the fully formatted problem. A Skolem sequence of order n is a sequence (s1, s2,...,s2n) of 2n integers satisfying the conditions: i) for every k in {1,2,3...,n} there exist exactly two elements si and sj with si = sj = k and ii) if si = sj = k with i ...continues

Combinatorial Result about the Binomial Coefficients

Please see the attached PDF file for the fully formatted problem. By the Binomial Theorem .... Therefore .... Evaluate the integral to get .... by a similar line of reasoning. Since this is an analysis problem, please be sure to be rigorous, and include as much detail as possible so that I can understand. Please also ...continues