Q.2 Two wizards try to create a spell using all the four elements, water, air, fire, and
earth. For this, they decide to mix all these elements in all possible orders. They
also decide to work independently. After trying all possible combination of
elements, they conclude that the spell does not work.
How many attempts does each wizard make before coming to this conclusion,
independently?
(A) 24
(B) 48
(C) 16
(D) 12
Each wizard makes 24 attempts independently before concluding the spell does not work. The four distinct elements (water, air, fire, earth) must be arranged in all possible unique sequences, which is a permutations problem.
Problem Analysis
The wizards mix all four elements in all possible orders, meaning every element is used exactly once in each attempt. Since the elements are distinct and order matters, calculate the number of permutations of 4 items: 4!=4×3×2×1=24. They work independently, so each tries all 24 sequences.
Option Breakdown
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(A) 24: Correct. Matches 4!for distinct permutations without repetition.
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(B) 48: Incorrect. Possibly double-counts (e.g., both wizards combined), but question asks per wizard.
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(C) 16: Incorrect. Equals 42 (repetitions allowed, like 2 positions), not applicable here.
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(D) 12: Incorrect. Could be 4×3(permutations of 2 from 4) or other partial counts.
Water, air, fire, and earth—the classic four elements—form the heart of this wizards four elements permutations puzzle from competitive exams like GATE CSE. Two wizards independently mix these distinct elements in all possible orders to create a spell, trying every unique sequence before concluding it fails. This tests fundamental permutation concepts crucial for CSIR NET Life Sciences and math sections.
Core Calculation
For n distinct objects arranged where order matters and none repeat, the count is n! (factorial). Here, :
4!=4×3×2×1=24
. Each wizard tests these 24 sequences independently—no sharing, no repetitions.
List of First Few Permutations (Water=W, Air=A, Fire=F, Earth=E):
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W-A-F-E
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W-A-E-F
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W-F-A-E
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A-W-F-E
(Full 24 via systematic arrangement).
Why Not Combinations?
The problem specifies orders, not selections—ruling out combinations (C(4,4)=1). Repetitions (e.g., 44=256) don’t fit “all the four elements” exactly once.
Exam Relevance
Appears in GATE CSE 2024 Set 2 GA. Builds permutation skills for biology/genetics (e.g., sequencing DNA bases) and biotech applications in CSIR NET.
Answer: (A) 24—each wizard’s independent attempts.
