Explanation
1. Condition for Failure:
Total number of papers = 9.
A candidate is successful if: Number of passes > Number of fails.
A candidate is unsuccessful if: Number of fails ≥ Number of passes.
2. Identifying Unsuccessful Cases:
Let r be the number of papers the candidate fails.
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If r=5, fails = 5, passes = 4 (Unsuccessful)
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If r=6, fails = 6, passes = 3 (Unsuccessful)
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If r=7, fails = 7, passes = 2 (Unsuccessful)
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If r=8, fails = 8, passes = 1 (Unsuccessful)
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If r=9, fails = 9, passes = 0 (Unsuccessful)
3. Calculating the Number of Ways:
The number of ways to fail in exactly r papers out of 9 is 9Cr.
Total unsuccessful ways = 9C5+9C6+9C7+9C8+9C9
4. Using Binomial Properties:
We know that the sum of all combinations is:
Also, by symmetry (nCr=nCn−r):
9C0+9C1+9C2+9C3+9C4=9C9+9C8+9C7+9C6+9C5
Let the sum of unsuccessful ways be S. Since there are 10 terms in total and the sum is split exactly in half:
Correct Option: 2
