4. Consider the following identity: n 2 2n Σ (3) - (30) n (1) Perhaps it seems mysterious? But if you try a few small values of n, you may begin to suspect it is correct. (a) Suppose n N and that we have a set S of cardinality 2n containing n distinct objects, all coloured red, and n distinct objects, all coloured blue. Let ke Z satisfy 0 ≤ k
4. Consider the following identity: n 2 2n Σ (3) - (30) n (1) Perhaps it seems mysterious? But if you try a few small values of n, you may begin to suspect it is correct. (a) Suppose n N and that we have a set S of cardinality 2n containing n distinct objects, all coloured red, and n distinct objects, all coloured blue. Let ke Z satisfy 0 ≤ k
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:4. Consider the following identity:
2
Σ(3) - (30)
(2n).
(1)
Perhaps it seems mysterious? But if you try a few small values of n, you may begin to suspect it is
correct.
(a) Suppose n E N and that we have a set S of cardinality 2n containing n distinct objects, all
coloured red, and n distinct objects, all coloured blue. Let k € Z satisfy 0 <k <n. How many
ways are there to sample n objects from S, subject to k of them being red? Briefly justify your
answer.
(b) Using your solution to (a), prove identity (1). (If you have the correct answer in part (a), this is
not a lengthy proof!)
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