Chapter 2. Number of Cases
Higher category: 【Statistics】 Statistics Overview
1. overview
2. sequence
3. combination
a. Example problems for number of cases
1. Overview
⑴ sampling
① with-replacement: re-injecting and extracting something already extracted
② without-replacement: extracting without re-injecting what has already been extracted.
⑵ types of number of cases
Figure 1. type of number of cases
2. Permutation
⑴ definition: the number of cases in the order of k balls among n balls. consideration of order. without-replacement
⑵ permutation with repetition: consideration of order. with-replacement
⑶ permutation of multisets: permutation with something that are identical
⑷ circular permutation: number of cases sitting around a round table
3. Combination (binary coefficient)
⑴ Overview
① definition: number of cases of combination of k balls among n balls. no consideration of order. without-replacement
② That is, choosing r items from n distinct items without considering the order and without repetition.
③ Examples: cases of selection, choosing representatives (roles that are the same).
⑵ binary coefficient may be expressed in a triangule of Pascal
Figure 2. binary coefficient and triangle of Pascal
⑶ combination with repetition
① definition: consideration of order. with-replacement
nHk
② equivalent expression
a1 + ··· + an = k, = k, ai ≥ 0
⇔ A1 + ··· + An = k+n, Ai ≥1
⇔ □ | □ | ··· | □, # of □ = (k+n) and # of | = (k+n-1)
⇔ nHk = n+k-1Cn-1 = n+k-1Ck
⑷ Formula 1. The number of ways to choose r items from n items is always the same as the number of ways to choose the n-r items that are not selected.
⑸ Formula 2.
⑹ Formula 3.
⑺ Formal 4.
① Combinatoric interpretation
⑻ Formula 5.
① Algebraic interpretation
② Combinatoric interpretation
○ nCk: number of cases in which k numbers are selected out of n numbers
○ n-1Ck: number of cases in which k numbers are selected excluding a specific number.
○ n-1Ck-1: number of cases in which k numbers are selected including a specific number
⑼ Formula 6
① algebraic interpretation
② Combinatoric interpretation
○ situation: number of cases in which n+1 numbers are selected from 1 to n+m+1 numbers
○ nCn: number of cases in which n+1 is the largest number of a combination
○ n+1Cn: number of cases in which n+2 is the largest number of a combination
○ n+mCn: number of cases in which n+m+1 is the largest number of a combination
⑽ Formula 7
① Algebraic interpretation
② Combinatoric interpretation
○ the probability of tossing a coin until the front or the back comes out n+1 times = 1
○ nCn × (½)n+1 × 2: the probability that the same side as the last side comes out n times, and the different side comes out 0 times
○ n+1Cn × (½)n+2 × 2: the probability that the same side as the last side will come out n times, and the different side will come out 1 time
○ 2nCn × (½)2n+1 × 2: the probability that the same side as the last side comes out n times, and the different side comes out n times
Input: 2019.06.27 09:48