Problem 10
How many cards, from a regular deck, do we need to pick up in order to guarantee that at least 3 of them are of the same suit?
Problem 11
Assume that in a group of six people, each pair of two people consists either of 2 friends or 2 enemies. Show that there are either 3 mutual friends or 3 mutual enemies in the group.
Permutations and Combinations
Problem 12
From a class of 20 students, how many ways are there to form a committee of 3 students with named positions (President, VP, Secretary)?
Permutations
Definition
An k-permutation of a set with n elements is an ordered arrangement of k elements from that set. The number of such k-permutations is denoted P(n,k), or sometimes nPk. An n-permutation is simply called a permutation.
Problem 13
How many permutations of the set S = {1, 2, 3, 4, 5} are there?
Problem 14
From a class of 20 students, how many ways are there to form a committee of 3 students without named positions?
Combinations Definition
An k-combination of a set with n elements is k-elements subset from that set. The number of such k-combinations is denoted C(n,k), or sometimes nCk or kn (also
called binomial coefficient).
Problem 15
A chocolate box contains chocolates in 7 flavours: black, white, cherry, milk, nuts, orange and truffles. Assuming that there are at least 4 of each, in how many different ways can we pick 4 chocolates? (The order does not count)
Problem16 n n Prove the following combinatorial identity: k = n k .
Problem17 n+1 n n Give combinatorial proof of Pascals Identity: k = k 1 + k .
Pascals Triangle
n+1 n n k =k1+k
n = n = 1 0 n
20 21 2 3 3 3 3
0 101
40 41 42 43 4
50 51 52 53 54 5
60 61 62 63 64 65 6
70 71 72 73 74 75 76 7
80 81 82 83 84 85 86 87 8 90 91 92 93 94 95 96 97 98 9
100 101 102 103 104 105 106 107 108 109 10 0 1 2 3 4 5 6 7 8 9 10
Binomial Theorem
(x+y)2 = (x + y)3 = (x + y)4 =
.
(x + y)n =
= n0xny0 +n1xn1y +n2xn2y2 ++nx0yn
x2+2xy+y2
x3 +3x2y +3xy2 +y3
x4 +4x3y +6x2y2 +4xy3 +y4
n knxnkyk
k=0
Problem 18
What is the coefficient of x12y13 in the expression of (2x + y)25?
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