There are n white and n black balls marked 1,2, ⋯, n. The number of ways in which we can arrange

step 1 In this problem, you have to arrange n white and n black balls such that the neighboring balls a

There are n white and n black balls marked 1,2, ⋯, n. The...

There are $n$ white and $n$ black balls marked 1,2, $\cdots, n$. The number of ways in which we can arrange these balts in a row so that neighbouring balls are of different colours is
(a) $n !$
(b) $(2 n) !$
(c) $2(n)^{2}$
(d) $\frac{(2 n) !}{(n !)^{2}}$.

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Counting with Constraints

Counting with constraints involves determining the number of valid arrangements of objects when certain conditions or restrictions have to be met. This technique is essential in problems where not all theoretically possible arrangements are allowed, due to imposed conditions like alternating colors or types.

Factorial Function

The factorial function, denoted by ‘n!’, represents the product of all positive integers up to n and is a fundamental building block in combinatorial mathematics. It is used extensively in permutation problems to count the number of ways objects can be ordered.

Alternating Arrangements

Alternating arrangements are a specific type of permutation where objects of different categories must be arranged in an alternating pattern. This constraint forces a structured ordering, impacting how many valid arrangements exist and is often solved by partitioning the problem into separate cases based on the starting element.

Permutations

Permutations refer to the different arrangements of a set of distinct objects in a particular order. In combinatorial problems, permutations are calculated using factorial expressions and are crucial when the order of objects is significant.

Transcript

Please give Ace some feedback