Combinations Identity
Using the standard combinatorial identity, the value of the sum C(8,0) + C(8,1) + C(8,2) + ... + C(8,8), summing all binomial coefficients of order 8, equals which value?
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About This Question
- Subject
- mathematics
- Chapter
- permutations and combinations
- Topic
- combinations identity
- Difficulty
- Easy
- Year
- 2025
Solution
Correct Answer:
256
The sum of all binomial coefficients of a given order equals a power of two, a direct consequence of the binomial theorem central to JEE Advanced. Setting x = 1 in the expansion of (1 + x)^n gives the sum of C(n,k) for k from 0 to n equal to 2^n. For n = 8, this sum is 2^8 = 256. Combinatorially, the sum counts all subsets of an 8-element set, since C(8,k) counts subsets of size k, and the total number of subsets is 2^8. Option 128 = 2^7 uses the wrong exponent. Option 64 = 2^6 is too small. Option 512 = 2^9 is too large. Hence the sum is 256. Plausibility check: the identity says summing a full row of Pascal's triangle doubles the previous row's sum, and row 8 totals 2^8 = 256, matching both the subset count and the binomial substitution.
This easy difficulty mathematics question is from the chapter permutations and combinations, covering the topic of combinations identity. It appeared in the 2025 exam.
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