Given 2n objects, number of ways to select n objects

  • Thread starter Thread starter nowimpsbball
  • Start date Start date
Click For Summary
SUMMARY

The discussion focuses on calculating the number of ways to select n objects from a set of 2N objects, consisting of N identical objects and N distinct objects. The participant proposes using combinatorial formulas, specifically C(n, k) for distinct selections and a modified approach for identical objects. The conclusion drawn is that the total number of combinations can be expressed as the sum of selections from both sets, leading to the formula 2^n for the total ways to select n objects.

PREREQUISITES
  • Understanding of combinatorial mathematics, specifically combinations and permutations.
  • Familiarity with the concepts of identical and distinct objects in set theory.
  • Knowledge of basic algebraic manipulation and summation notation.
  • Experience with mathematical proofs and deriving formulas.
NEXT STEPS
  • Study the principles of combinatorial counting, focusing on combinations and permutations.
  • Learn about the binomial theorem and its applications in combinatorial problems.
  • Explore advanced topics in set theory, particularly the handling of identical versus distinct elements.
  • Investigate generating functions as a method for solving combinatorial problems.
USEFUL FOR

Students and educators in mathematics, particularly those studying combinatorics, as well as anyone interested in solving complex selection problems involving identical and distinct objects.

nowimpsbball
Messages
13
Reaction score
0

Homework Statement


Given N identical objects and N additional objects that are different from these and from each other, find the number of ways to select n objects out of these 2N objects.


Homework Equations


Either P(n,k) or C(n,k) or n^k or maybe even (n+k+1)/k


The Attempt at a Solution


Looking at these...so half the set is identical and the other half is distinct from the first half and from them selves. So I will call the first set S and the second set T to help me keep them apart.
So for the first set everything is identical and w can select up to N of them... my notes seem to indicate i would go about it as (S+N+1)/N and then the second set they are distinct from each other and we can select N of them, so T^N? Are those two formulas in the right direction? If so would I multiply those together to get the correct answer?
 
Physics news on Phys.org
Just count them. Number of ways to select n of A, 0 of B is 1.
n-1 of A, 1 of B is n
n-2 of A, 2 of B is n(n-1), etc.

So you get sum{i=0..n}(n choose i) = 2^n.
 

Similar threads

Prove that ##\lim\limits_{x \to \infty} f(x) = 0##
  • · Replies 3 ·
Replies
3
Views
2K
How many subsets of size three or less are in a n-object set
  • · Replies 4 ·
Replies
4
Views
2K
Finding the nth roots of a complex number
  • · Replies 22 ·
Replies
22
Views
2K
Prove ##|\{ q\in\mathbb{Q}: q>0 \} |=|\mathbb{N}|##.
  • · Replies 3 ·
Replies
3
Views
2K
Prove if ##a\cdot c = b \cdot c## then ##a = b## using Peano postulates
  • · Replies 2 ·
Replies
2
Views
1K
Proving that ##\lim [\sqrt{4n^2 +n} - 2n] = \frac{1}{4}##
  • · Replies 9 ·
Replies
9
Views
3K
Undergrad Understanding permutations and combinations in a coin toss experiment
  • · Replies 6 ·
Replies
6
Views
2K
General formula for a combination of four categories
  • · Replies 3 ·
Replies
3
Views
2K
Prove ##a \cdot 1 = a = 1 \cdot a## for ##a \in \mathbb{N}##
  • · Replies 1 ·
Replies
1
Views
2K
Prove ##1 + a=s(a)=a+1## for ##a \in \mathbb{N}##
  • · Replies 1 ·
Replies
1
Views
2K