{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

05_Counting-1_packed

# Combina8ons combina8ons k element subsets

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: , {B} , {C} ⎛ 3⎞ number of two-element subsets = ⎜ =3 2⎟ ⎝ ⎠ {A, B} , {A, C} , {B, C} Example: a set with 3 elements ⎛ 3⎞ number of zero-element subsets = ⎜ ⎟ ≡ 1---just the empty set ⎝0⎠ A Set S = B C ⎛ 3⎞ number of one-element subsets = ⎜ ⎟=3 ⎝1⎠ {A} , {B} , {C} ⎛ 3⎞ number of two-element subsets = ⎜ =3 2⎟ ⎝ ⎠ {A, B} , {A, C} , {B, C} ⎛ 3⎞ number of three-element subsets = ⎜ = 1---just the set S = { A, B, C } itself 3⎟ ⎝ ⎠ ⎛ 3⎞ ⎛ 3⎞ ⎛ 3⎞ ⎛ 3⎞ total number of subsets = ⎜ +⎜ +⎜ +⎜ = 1 + 3 + 3 + 1 = 8 = 23 ⎝ 0⎟ ⎝ 1⎟ ⎝ 2⎟ ⎝ 3⎟ ⎠ ⎠ ⎠ ⎠ + b (a + b) = (a + ba ⋅(+ )( b) … ⋅ a ) n n times ⎛ n ⎞ n −1 ⎛ n ⎞ n − 2 2 ⎛ n ⎞ 2 n − 2 ⎛ n ⎞ n −1 = a +⎜ a b+⎜ a b + … + ⎜ ⎟a b + ⎜ ⎟ ab + b n ⎟ ⎟ ⎝ n − 1⎠ ⎝ n − 2⎠ ⎝ 2⎠ ⎝ 1⎠ n Coefficient of a k = number of ways we can choose k a' s from the n terms in parentheses ⎛ n⎞ Convention : ⎜ ⎟ = 1 ⎝ 0⎠ ⎛ n⎞ k n− k n ⎛ n⎞ For example, 2 = (1 + 1) = ∑⎜ ⎟ ⋅ 1 ⋅ 1 = ∑⎜ ⎟ k k k = 0⎝ ⎠ k = 0⎝ ⎠ n n n Ilya Pollak Pascal’s triangle Ilya Pollak Pascal’s triangle Ilya Pollak Pascal’s triangle Ilya Pollak Pascal’s triangle: Deriva8on ⎛ n ⎞ ⎛ n⎞ n! n! + ⎜ ⎟+⎜ ⎟= k − 1⎠ ⎝ k ⎠ ( k − 1)!( n − ( k − 1))! k!( n − k )! ⎝ k ⋅ n! ( n − k + 1) ⋅ n! = + k k − ( n − k + 1)! k!( n k( nk + ⋅ 1)! )! ( − ⋅ − 1) = k! ( n − k +1)! ( k + n − k + 1) ⋅ n! k!( n − k + 1)! ( n + 1) ⋅ n! = k!( n − k + 1)! ( n + 1)! = k!( n − k + 1)! ⎛ n + 1⎞ =⎜ ⎟ k⎠ ⎝ = Ilya Pollak...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online