Mastering Basic Combination Identities: Foundations & Core Skills

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| Questions: 15 | Updated: Dec 1, 2025
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1) The number of combinations of n objects taken r at a time is given by:

Explanation

By definition, binom(n,r)=n!/(r!(n-r)!).

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About This Quiz
Mastering Basic Combination Identities: Foundations & Core Skills - Quiz

Are you ready to strengthen your understanding of how combinations work? This quiz walks you through the essential identities behind binomial coefficients — from symmetry and Pascal’s rule to subset counting and the binomial expansion. You’ll compute values like (n choose r), explore the meaning behind choosing k items, and... see moreapply classic identities such as ∑(n choose k) = 2ⁿ. Along the way, you’ll connect factorial formulas with combinatorial reasoning, practice quick evaluations, and see how identities appear naturally in counting problems. By the end, you’ll feel confident using these tools to simplify expressions and solve combination-based questions with ease. see less

2)
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2) Compute ((5¦2)).

Explanation

(5 choose 2)=10.

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3) What are the values of ((7¦0)) and ((7¦7))?

Explanation

Both equal 1.

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4) For every integer (n ≥ 1), we have ((n¦1)=n).

Explanation

Choosing 1 object from n yields n.

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5) Using symmetry identity, ((10¦3) = (10¦7)). What is this value?

Explanation

(10 choose 3)=120.

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6) Use Pascal’s identity to compute ((5¦2)+(5¦3)).

Explanation

Pascal’s identity.

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7) Compute ((4¦0)+(4¦1)+(4¦2)+(4¦3)+(4¦4)).

Explanation

Sum of row =16.

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8) Sum_{k=0}^n (n¦k)=2^n counts all subsets.

Explanation

2^n subsets.

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9) Which is Pascal’s identity?

Explanation

Correct Pascal rule.

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10) Use hockey-stick: ((3¦3)+(4¦3)+(5¦3)).

Explanation

Sum=15.

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11) (n¦k-1)+(n¦k)=(n+1¦k) is Pascal’s form.

Explanation

Shifted Pascal.

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12) Compute ((8¦2)).

Explanation

(8*7)/(2*1)=28.

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13) Coefficient of x^3 y^2 in (x+y)^5?

Explanation

Choose positions of y’s.

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14) Sum_{k} (-1)^k (n¦k)=0 for n>=1.

Explanation

Expansion of (1-1)^n.

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15) Meaning of ((6¦2))?

Explanation

Counts 2-element subsets.

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The number of combinations of n objects taken r at a time is given by:
Compute ((5¦2)).
What are the values of ((7¦0)) and ((7¦7))?
For every integer (n ≥ 1), we have ((n¦1)=n).
Using symmetry identity, ((10¦3) = (10¦7)). What is this value?
Use Pascal’s identity to compute ((5¦2)+(5¦3)).
Compute ((4¦0)+(4¦1)+(4¦2)+(4¦3)+(4¦4)).
Sum_{k=0}^n (n¦k)=2^n counts all subsets.
Which is Pascal’s identity?
Use hockey-stick: ((3¦3)+(4¦3)+(5¦3)).
(n¦k-1)+(n¦k)=(n+1¦k) is Pascal’s form.
Compute ((8¦2)).
Coefficient of x^3 y^2 in (x+y)^5?
Sum_{k} (-1)^k (n¦k)=0 for n>=1.
Meaning of ((6¦2))?
Alert!

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