Advanced Techniques in Combination Identities and Binomial Structures

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Ekaterina Yukhnovich, PhD |
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Ekaterina V. is a physicist and mathematics expert with a PhD in Physics and Mathematics and extensive experience working with advanced secondary and undergraduate-level content. She specializes in combinatorics, applied mathematics, and scientific writing, with a strong focus on accuracy and academic rigor.
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| Questions: 15 | Updated: Dec 12, 2025
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1) In the binomial expansion ((x+y)^n), the coefficient of (x^k y^{n-k}) is:

Explanation

The binomial theorem states ((x+y)^n = sum...).

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About This Quiz
Advanced Techniques In Combination Identities and Binomial Structures - Quiz

Ready to explore the deeper structure behind binomial coefficients? This postgraduate-level quiz challenges you with powerful identities used in advanced combinatorics, including Vandermonde’s identity, the hockey-stick pattern, squared-binomial sums, and binomial expansions with variable coefficients. You’ll analyze how combinations split across groups, apply symmetry to compute large coefficients, and use... see morethe binomial theorem to evaluate algebraic expressions efficiently. Each question pushes your understanding of why these identities work — not just how to apply them. By the end, you’ll have a sharper, more elegant grasp of combination identities and their role in higher-level counting arguments.
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2)
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2) Use the symmetry identity to compute ((7¦4)) via ((7¦3)).

Explanation

7 choose 4 = 35

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3) Which identity is obtained by splitting all k-subsets of an n-element set according to whether they contain a fixed element?

Explanation

Pascal identity

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4) Evaluate sum (j=2 to 6) (j¦2)

Explanation

Hockey stick gives (7 choose 3)

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5) Compute sum (4¦k)2^k

Explanation

(1+2)^4

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6) Identity holds? (n¦k)=n/(n-k)(n-1¦k)

Explanation

From factorial manipulation

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7) ((n¦k)) always integer?

Explanation

Counts subsets

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8) Sum (3¦k)^2

Explanation

6 choose 3

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9) Even binomial sum ((8¦0)+(8¦2)+...)

Explanation

2^7=128

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10) Vandermonde identity

Explanation

Vandermonde

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11) What is (5¦7)?

Explanation

Can't choose 7 from 5

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12) Simplify (n¦0)+(n¦1)

Explanation

1+n

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13) 3-person committees from 11

Explanation

11 choose 3

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14) Pascal’s triangle property?

Explanation

Definition

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15) Largest binomial coefficient location

Explanation

Unimodal centered

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Ekaterina Yukhnovich |PhD |
College Expert
Ekaterina V. is a physicist and mathematics expert with a PhD in Physics and Mathematics and extensive experience working with advanced secondary and undergraduate-level content. She specializes in combinatorics, applied mathematics, and scientific writing, with a strong focus on accuracy and academic rigor.
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In the binomial expansion ((x+y)^n), the coefficient of (x^k y^{n-k})...
Use the symmetry identity to compute ((7¦4)) via ((7¦3)).
Which identity is obtained by splitting all k-subsets of an n-element...
Evaluate sum (j=2 to 6) (j¦2)
Compute sum (4¦k)2^k
Identity holds? (n¦k)=n/(n-k)(n-1¦k)
((n¦k)) always integer?
Sum (3¦k)^2
Even binomial sum ((8¦0)+(8¦2)+...)
Vandermonde identity
What is (5¦7)?
Simplify (n¦0)+(n¦1)
3-person committees from 11
Pascal’s triangle property?
Largest binomial coefficient location
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