There are n persons (n ≥3), among whom are A and B, who are made to stand in a row in random ord

step 1 in this problem note that the person that must stand between a and b can be chosen in n minus tw

There are n persons (n ≥3), among whom are A and B, who are...

There are $n$ persons $(n \geq 3)$, among whom are $A$ and $B$, who are made to stand in a row in random order. Probability that there is exactly one person between $A$ and $B$ is
(A) $\frac{n-2}{n(n-1)}$
(B) $\frac{2(n-2)}{n(n-1)}$
(C) $2 / n$
(D) none os these

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Key Concepts

Probability Calculation

Probability calculation in discrete sample spaces involves determining the ratio of favorable outcomes to the total number of outcomes. It is essential to accurately calculate both counts using combinatorial methods to determine the probability of events subject to specific conditions.

Combinatorial Counting

This concept involves methods to count the number of favorable outcomes that satisfy certain conditions, such as the specific positions of objects or persons. By employing combinatorial counting techniques, one can determine the number of arrangements where specific criteria, like spacing constraints, are met.

Permutations

Permutations refer to the different ways of arranging a set of items in a specific order. In problems involving arrangements of persons, the total number of permutations is calculated by factorials. Understanding permutations is fundamental when calculating the total number of possible arrangements.

Spacing Constraints in Arrangements

This entails applying restrictions on the positions of items relative to each other, such as requiring a fixed number of items to be between two specified items in an arrangement. Recognizing and enforcing these constraints is crucial when counting special cases in permutation problems.

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