Pascal’s Triangle Practice Test

Advanced Algebra Practice Test: ACT math skills.

Pascal’s Triangle Practice Test

This test has 20 questions

Instant feedback · Worked explanations

Pascal's Triangle · Coefficient Courtyard

Every tile is built from the two above.

Pascal's Triangle is both a number pattern and a storage system for binomial coefficients. Its edges, symmetry, diagonals, and row sums provide reliable ways to construct entries and check expansion coefficients.

Review mapConstruct rows, locate entries, connect rows to binomial powers, use symmetry, inspect row sums, read diagonal patterns, and avoid row-numbering errors.

Start at row zeroThe top entry belongs to the zeroth row.

Keep edge onesEvery row begins and ends with one.

Add parentsInterior entries use the two above.

Use symmetryMatching positions from both ends agree.

Match powersRow number equals binomial exponent.

Build the triangle from three local rules

No long formula is required when one row is constructed directly from the row above.

Terrace 01 · Foundation

The top is one

1

This is row zero and represents the zeroth binomial power.

Edges stay one

1 \dots 1

Place one at both ends before computing interior entries.

Interior entries add parents

c = p + q

Each interior tile is the sum of the two closest entries above it.

Alignment matters: rows are offset so an interior entry sits between its two parents. Adding vertically aligned entries instead produces the wrong row.

The parent-sum rule creates the next row

Work from left to right after placing the two edge entries.

Terrace 02 · Construction

Worked row transition

1 4 6 4 1

Add adjacent pairs while preserving edge ones.

1 (1 + 4) (4 + 6) (6 + 4) (4 + 1) 1

1 5 10 10 5 1

Read the early rows as a numbered reference

Row numbering begins at zero, not one.

Terrace 03 · Row map

\begin{matrix} Row 0: 1 \\ Row 1: 1 1 \\ Row 2: 1 2 1 \\ Row 3: 1 3 3 1 \\ Row 4: 1 4 6 4 1 \\ Row 5: 1 5 10 10 5 1 \\ Row 6: 1 6 15 20 15 6 1 \end{matrix}

Entry count

entries in row = n + 1

Center behavior

An even-numbered row has one central entry. An odd-numbered row has two equal central entries.

Symmetry check

Reading any row backward gives the same sequence.

Each row supplies one binomial expansion

The row number matches the outside exponent, and row entries become coefficients.

Terrace 04 · Algebra link

{(a + b)}^{5} = a^{5} + 5 a^{4} b + 10 a^{3} b^{2} + 10 a^{2} b^{3} + 5 a b^{4} + b^{5}

Coefficient row

1, 5, 10, 10, 5, 1

First exponent

5, 4, 3, 2, 1, 0

Second exponent

0, 1, 2, 3, 4, 5

Degree check

(5 - r) + r = 5

Combination notation gives every tile an address

The row number and zero-based position identify a Pascal entry.

Terrace 05 · Coordinates

entry at row n and position r = (\binom{n}{r})

Question

MathML setup

Result

Third entry of row eight

third entry \Rightarrow r = 2

(\binom{8}{2}) = 28

Fifth entry of row eight

fifth entry \Rightarrow r = 4

(\binom{8}{4}) = 70

Last entry of row eight

last position \Rightarrow r = 8

(\binom{8}{8}) = 1

Position warning: ordinary language counts the first entry as first, but the combination index counts that same entry from zero.

Symmetry cuts the work in half

Entries equally far from the two edges are identical.

Terrace 06 · Mirror axis

Combination identity

(\binom{n}{r}) = (\binom{n}{n - r})

Choosing a position from the left matches the corresponding position from the right.

Row example

1, 7, 21, 35, 35, 21, 7, 1

Pair the endpoints, then move inward to verify the mirror pattern.

Useful strategy: if a test asks for an entry near the right edge, its mirrored entry near the left edge may use a much smaller and easier index.

Row sums follow powers of two

This pattern comes from evaluating a binomial with both bases equal to one.

Terrace 07 · Sum fountain

\sum_{r = 0}^{n} (\binom{n}{r}) = 2^{n}

Row three

1 + 3 + 3 + 1 = 8 = 2^{3}

Row four

1 + 4 + 6 + 4 + 1 = 16 = 2^{4}

Row five

1 + 5 + 10 + 10 + 5 + 1 = 32 = 2^{5}

Fast check

If a completed row does not add to the matching power of two, at least one entry is incorrect.

Diagonal paths reveal familiar sequences

Reading shallow diagonals provides another construction and recognition tool.

Terrace 08 · Pathways

Three early patterns

edge: 1, 1, 1, 1, \dots

next diagonal: 1, 2, 3, 4, \dots

third diagonal: 1, 3, 6, 10, \dots

The third sequence contains triangular numbers formed by cumulative counting.

Use neighboring structure to find a missing entry

A missing tile can often be recovered locally without rebuilding the whole triangle.

Terrace 09 · Restoration

Known parents

p = 15, q = 20

Missing child

c = p + q = 15 + 20 = 35

Mirror check

If the missing position has a visible symmetric partner, both entries must agree.

Reverse reasoning: if a child and one parent are known, subtract the known parent from the child to recover the other parent. Confirm that the result also fits the row's symmetry.

Skills Covered

The test combines pattern construction, algebraic interpretation, and efficient checking.

Terrace 10 · Skills

Construct rowsPreserve edge ones and add adjacent parent entries.

Use row numberingMatch row zero through higher rows to binomial exponents.

Locate coefficientsConvert an entry position into a zero-based combination index.

Expand binomialsPair a Pascal row with decreasing and increasing powers.

Recognize patternsUse symmetry, row sums, edges, centers, and diagonals.

Recover missing entriesApply local addition, subtraction, and mirror checks.

How to Approach the Test

Identify whether the question is about construction, location, expansion, or a pattern.

Terrace 11 · Method

Read the row convention

Confirm that the top is labeled row zero.

Avoid a one-row shift.

Mark edges and parents

Keep edge ones fixed and identify the two entries above each interior position.

Use the local geometry.

Translate the requested position

Convert ordinary entry counting to a zero-based index when needed.

First entry uses index zero.

Choose the shortest rule

Use addition, symmetry, combination notation, or a known row.

Do not rebuild unnecessarily.

Connect to algebra carefully

Use the correct row and align coefficients with the power pattern.

Row number equals exponent.

Check the result

Inspect symmetry, edge ones, entry count, and row sum.

Patterns expose mistakes quickly.

Common Mistakes

Most errors come from row numbering, alignment, or confusing entry position with combination index.

Terrace 12 · Repairs

Calling the top row one

The standard binomial convention labels the top as row zero.

Forgetting edge ones

Every completed row must begin and end with one.

Adding the wrong parents

An interior tile uses the two diagonally adjacent entries above.

Using entry number as the index

The first entry corresponds to index zero, not one.

Reading the wrong binomial row

The outside exponent selects the row directly.

Ignoring symmetry

Unequal mirrored positions signal a construction error.

Adding a row incorrectly

The row sum must equal the matching power of two.

Choosing an intermediate value

Return the requested entry, coefficient, row, or expansion.

Final Pascal audit

Verify the local construction rule and the global row patterns.

Final terrace · Verified

Mosaic Row Check

A correct row works both locally, through parent sums, and globally, through symmetry and row totals.

Edges · Parents · Mirror · Sum

Was the top treated as row zero?Row numbering must match binomial exponents.

Does the row begin and end with one?The edge entries never change.

Does every interior entry add its parents?Use the two diagonally adjacent entries above.

Is the row symmetric?Matching distances from the edges must agree.

Is the entry count correct?Row number plus one gives the number of entries.

Does the row sum fit its power of two?Use the total as a final arithmetic check.

Use this free 20-question practice test for high school Advanced Algebra review, ACT-style skill practice, placement preparation, or classroom practice. You can retake the test without creating an account. The examples in this review block are illustrative and are not copies of the test questions.