Mastering Basic Combination Identities: Foundations & Core Skills

1) The number of combinations of n objects taken r at a time is given by:

N!/((n-r)!)

N!/r!

N!/(r!(n-r)!)

N^r

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

Explanation

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

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) Compute ((5¦2)).

(5 choose 2)=10.

Explanation

(5 choose 2)=10.

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

0 and 0

1 and 1

7 and 7

0 and 1

Both equal 1.

Explanation

Both equal 1.

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

True

False

Choosing 1 object from n yields n.

Explanation

Choosing 1 object from n yields n.

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

120

720

(10 choose 3)=120.

Explanation

(10 choose 3)=120.

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

(6¦2)

(6¦3)

(6¦4)

(5¦4)

Pascal’s identity.

Explanation

Pascal’s identity.

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

Sum of row =16.

Explanation

Sum of row =16.

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

True

False

2^n subsets.

Explanation

2^n subsets.

9) Which is Pascal’s identity?

(n¦k)=(n¦n-k)

(n¦k)+(n¦k+1)=(n+1¦k+1)

Sum (n¦k)=2^n

(n¦k)=n/k (n-1¦k-1)

Correct Pascal rule.

Explanation

Correct Pascal rule.

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

(5¦4)

(6¦4)

(6¦3)

(7¦4)

Sum=15.

Explanation

Sum=15.

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

True

False

Shifted Pascal.

Explanation

Shifted Pascal.

12) Compute ((8¦2)).

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

Explanation

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

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

(5¦2)

(5¦3)

(5¦4)

(5¦1)

Choose positions of y’s.

Explanation

Choose positions of y’s.

14) Sum_{k} (-1)^k (n¦k)=0 for n>=1.

True

False

Expansion of (1-1)^n.

Explanation

Expansion of (1-1)^n.

15) Meaning of ((6¦2))?

Permutations of 6

Ordered pairs

Unordered pairs

All subsets

Counts 2-element subsets.

Explanation

Counts 2-element subsets.