Probability
Words/Arrangement Probability
nta_pyq_2025_apr
Grade 12
Question:
Let $S$ be the set of all the words that can be formed by arranging all the letters of the word GARDEN. From the set $S$, one word is selected at random. The probability that the selected word will NOT have vowels in alphabetical order is:
$\frac{1}{2}$
$\frac{1}{4}$
$\frac{2}{3}$
$\frac{1}{3}$
Step-by-Step Solution
Key Concept: Vowels in GARDEN are A and E. In any arrangement, the two vowels occupy 2 of the 6 positions; exactly half of all arrangements have A before E (alphabetical order).
GARDEN has vowels A, E. By symmetry, P(A before E) $= 1/2$. P(NOT in alphabetical order) $= 1 - \frac{1}{2} = \frac{1}{2}$.
Correct Answer: $\frac{1}{2}$