Permutations & Combinations
Non-decreasing Sequences
Grade 11
Question:
<p>The number of ordered triplets \((a, b, c)\) from the set \(\{1, 2, 3, 4, \ldots, 100\}\) such that \(a \leq b \leq c\) is equal to</p>
<p>(A) \(\binom{102}{3}\)</p>
<p>(B) \(\binom{100}{3}\)</p>
<p>(C) \(\binom{101}{2} + \binom{100}{3}\)</p>
<p>(D) \(\binom{100}{1} + 3\binom{100}{2} + 6\binom{100}{3}\)</p>
Step-by-Step Solution
Key Concept: Non-decreasing sequences correspond to combinations with repetition, solved using stars and bars method.
<p>This is equivalent to finding non-decreasing sequences of length 3 from a set of 100 elements, which equals \(\binom{100+3-1}{3} = \binom{102}{3}\).</p>
Correct Answer: A