Bell Numbers Online Test

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Bell Numbers

Description: Bell numbers are a sequence of integers that count the number of ways to partition a set of n elements into non-empty subsets.
Number of Questions: 14
Created by: Aliensbrain Bot
Tags: combinatorics bell numbers

What is the formula for the Bell number B(n)?

B(n) = (1/e) * sum(k! * S(n, k) for k from 0 to n)
B(n) = (1/e) * sum(k! * S(n, k) for k from 1 to n)
B(n) = (1/e) * sum(k! * S(n, k) for k from 2 to n)
B(n) = (1/e) * sum(k! * S(n, k) for k from 3 to n)

What is the Bell number for n = 3?

5
10
15
20

What is the Bell number for n = 4?

15
20
25
30

What is the Bell number for n = 5?

52
60
68
76

What is the Bell number for n = 6?

203
220
238
256

What is the Bell number for n = 7?

877
924
972
1020

What is the Bell number for n = 8?

4140
4320
4500
4680

What is the Bell number for n = 9?

21147
21900
22663
23436

What is the Bell number for n = 10?

115975
118750
121535
124320

What is the relationship between Bell numbers and Stirling numbers of the second kind?

B(n) = (1/e) * sum(k! * S(n, k) for k from 0 to n)
B(n) = (1/e) * sum(k! * S(n, k) for k from 1 to n)
B(n) = (1/e) * sum(k! * S(n, k) for k from 2 to n)
B(n) = (1/e) * sum(k! * S(n, k) for k from 3 to n)

What is the asymptotic formula for the Bell number B(n)?

B(n) ~ (1/sqrt(2 * pi * n)) * (n/e)^n
B(n) ~ (1/sqrt(2 * pi * n)) * ((n+1)/e)^(n+1)
B(n) ~ (1/sqrt(2 * pi * n)) * ((n-1)/e)^(n-1)
B(n) ~ (1/sqrt(2 * pi * n)) * ((n+2)/e)^(n+2)

What is the generating function for the Bell numbers?

exp(e^x - 1)
exp(e^x + 1)
exp(e^x - 2)
exp(e^x + 2)

What is the relationship between Bell numbers and exponential generating functions?

The exponential generating function for the Bell numbers is exp(e^x - 1).
The exponential generating function for the Bell numbers is exp(e^x + 1).
The exponential generating function for the Bell numbers is exp(e^x - 2).
The exponential generating function for the Bell numbers is exp(e^x + 2).

What is the relationship between Bell numbers and derangements?

The number of derangements of n elements is equal to the Bell number B(n).
The number of derangements of n elements is equal to the Bell number B(n+1).
The number of derangements of n elements is equal to the Bell number B(n-1).
The number of derangements of n elements is equal to the Bell number B(n+2).

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