Advanced Techniques in Combination Identities and Binomial Structures

Reviewed by Editorial Team
The ProProfs editorial team is comprised of experienced subject matter experts. They've collectively created over 10,000 quizzes and lessons, serving over 100 million users. Our team includes in-house content moderators and subject matter experts, as well as a global network of rigorously trained contributors. All adhere to our comprehensive editorial guidelines, ensuring the delivery of high-quality content.
Learn about Our Editorial Process
| By Thames
T
Thames
Community Contributor
Quizzes Created: 7387 | Total Attempts: 9,527,791
| Questions: 15 | Updated: Dec 1, 2025
Please wait...
Question 1 / 15
0 %
0/100
Score 0/100
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...).

Submit
Please wait...
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. see less

2)
You may optionally provide this to label your report, leaderboard, or certificate.
2) Use the symmetry identity to compute ((7¦4)) via ((7¦3)).

Explanation

7 choose 4 = 35

Submit
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

Submit
4) Evaluate sum (j=2 to 6) (j¦2)

Explanation

Hockey stick gives (7 choose 3)

Submit
5) Compute sum (4¦k)2^k

Explanation

(1+2)^4

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

Explanation

From factorial manipulation

Submit
7) ((n¦k)) always integer?

Explanation

Counts subsets

Submit
8) Sum (3¦k)^2

Explanation

6 choose 3

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

Explanation

2^7=128

Submit
10) Vandermonde identity

Explanation

Vandermonde

Submit
11) What is (5¦7)?

Explanation

Can't choose 7 from 5

Submit
12) Simplify (n¦0)+(n¦1)

Explanation

1+n

Submit
13) 3-person committees from 11

Explanation

11 choose 3

Submit
14) Pascal’s triangle property?

Explanation

Definition

Submit
15) Largest binomial coefficient location

Explanation

Unimodal centered

Submit
×
Saved
Thank you for your feedback!
View My Results
Cancel
  • All
    All (15)
  • Unanswered
    Unanswered ()
  • Answered
    Answered ()
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
Alert!

Advertisement