Permutations & Combinations
Selection with Restrictions
Grade 11
Question:
<p>Given \(n\) different objects arranged in a row. Then the number of ways of choosing \(k\) of them so that no two of them are consecutive is equal to</p>
<p>(A) \(\binom{n}{k}\)</p>
<p>(B) \(\binom{n-k}{k}\)</p>
<p>(C) \(\binom{n}{k} - (n-k+1)\)</p>
<p>(D) \(\binom{n-k+1}{k}\)</p>
Step-by-Step Solution
Key Concept: Choosing non-consecutive objects is equivalent to choosing k items from n-k+1 positions created by the remaining objects.
<p>If we choose k objects with no two consecutive from n objects, we can think of it as arranging k selected objects and (n-k) unselected objects such that between selected objects there is at least one unselected object. This gives \(\binom{n-k+1}{k}\).</p>
Correct Answer: D