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Question 1
What is the value of $(4 9)$ ?
No clue? Show me the answer
Correct answerCorrect!Incorrect
B126
Step-by-step walkthrough
Choose a solution method
Method #1Direct approach
Step 1: Write the formula
Use the binomial coefficient formula: $(4 9) = \frac{9 !}{4 ! \cdot 5 !}$
Step 2: Cancel common factors
Write out only the factors that don't cancel: $\frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1}$
Step 3: Compute numerator and denominator
Numerator: $9 \times 8 \times 7 \times 6 = 3024$ . Denominator: $4! = 24$ .
Step 4: Divide to get the answer
$\frac{3024}{24} = 126$ So $(4 9) = 126$ .
Method #2Process of Elimination
Step 1: Identify what is being asked
We need to compute $(4 9)$ using the formula $\frac{n !}{r ! ( n - r )!}$ with $n = 9$ , $r = 4$ .
Step 2: Eliminate 36
36 would equal $(2 9) = \frac{9 \times 8}{2} = 36$ , not $(4 9)$ . This is the wrong value of $r$ .
Step 3: Eliminate 84
84 equals $(3 9) = \frac{9 \times 8 \times 7}{6} = 84$ . Again, this uses $r = 3$ , not $r = 4$ .
Step 4: Eliminate 210
210 equals $(4 10)$ or $(6 10)$ , not $(4 9)$ . This is an off-by-one error in $n$ .
Step 5: Select the correct answer
126 is correct since $(4 9) = \frac{9 \times 8 \times 7 \times 6}{4 !} = \frac{3024}{24} = 126$ .
Question 2
Which row of Pascal's triangle gives the binomial coefficients for the expansion of $(a + b)^{6}$ ?
No clue? Show me the answer
Correct answerCorrect!Incorrect
B $1615201561$
Step-by-step walkthrough
Choose a solution method
Method #1Direct approach
Step 1: Identify the required row
The expansion of $(a + b)^{n}$ uses row $n$ of Pascal's triangle. For $(a + b)^{6}$ , we need row 6, which has 7 entries.
Step 2: Compute the entries using $\binom{6}{r}$
The entries are $(0 6), (1 6), \dots, (6 6)$ , which equal $1, 6, 15, 20, 15, 6, 1$ .
Step 3: Match to the correct option
The sequence $1, 6, 15, 20, 15, 6, 1$ matches the second option. Notice the symmetry: $(r 6) = (6 - r 6)$ .
Method #2Process of Elimination
Step 1: Identify what is being asked
We need the row of Pascal's triangle corresponding to $n = 6$ , which must have exactly $6 + 1 = 7$ entries.
Step 2: Eliminate the row with 6 entries
$15101051$ has only 6 entries — this is row 5, used for $(a + b)^{5}$ , not $(a + b)^{6}$ .
Step 3: Eliminate the row with 8 entries
$172135352171$ has 8 entries — this is row 7, used for $(a + b)^{7}$ .
Step 4: Eliminate the incorrect 7-entry row
$161281261$ has 7 entries but the values are wrong. For example, $(2 6) = 15 \neq = 12$ . These appear to be powers of 2, not binomial coefficients.
Step 5: Select the correct answer
$1615201561$ is row 6 of Pascal's triangle, confirmed by $(2 6) = 15$ and $(3 6) = 20$ .

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15 more questions in this topic

Free preview 2/17

Question 1
What is the value of $(4 9)$ ?
No clue? Show me the answer
Correct answerCorrect!Incorrect
B126
Step-by-step walkthrough
Choose a solution method
Method #1Direct approach
Step 1: Write the formula
Use the binomial coefficient formula: $(4 9) = \frac{9 !}{4 ! \cdot 5 !}$
Step 2: Cancel common factors
Write out only the factors that don't cancel: $\frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1}$
Step 3: Compute numerator and denominator
Numerator: $9 \times 8 \times 7 \times 6 = 3024$ . Denominator: $4! = 24$ .
Step 4: Divide to get the answer
$\frac{3024}{24} = 126$ So $(4 9) = 126$ .
Method #2Process of Elimination
Step 1: Identify what is being asked
We need to compute $(4 9)$ using the formula $\frac{n !}{r ! ( n - r )!}$ with $n = 9$ , $r = 4$ .
Step 2: Eliminate 36
36 would equal $(2 9) = \frac{9 \times 8}{2} = 36$ , not $(4 9)$ . This is the wrong value of $r$ .
Step 3: Eliminate 84
84 equals $(3 9) = \frac{9 \times 8 \times 7}{6} = 84$ . Again, this uses $r = 3$ , not $r = 4$ .
Step 4: Eliminate 210
210 equals $(4 10)$ or $(6 10)$ , not $(4 9)$ . This is an off-by-one error in $n$ .
Step 5: Select the correct answer
126 is correct since $(4 9) = \frac{9 \times 8 \times 7 \times 6}{4 !} = \frac{3024}{24} = 126$ .
Question 2
Which row of Pascal's triangle gives the binomial coefficients for the expansion of $(a + b)^{6}$ ?
No clue? Show me the answer
Correct answerCorrect!Incorrect
B $1615201561$
Step-by-step walkthrough
Choose a solution method
Method #1Direct approach
Step 1: Identify the required row
The expansion of $(a + b)^{n}$ uses row $n$ of Pascal's triangle. For $(a + b)^{6}$ , we need row 6, which has 7 entries.
Step 2: Compute the entries using $\binom{6}{r}$
The entries are $(0 6), (1 6), \dots, (6 6)$ , which equal $1, 6, 15, 20, 15, 6, 1$ .
Step 3: Match to the correct option
The sequence $1, 6, 15, 20, 15, 6, 1$ matches the second option. Notice the symmetry: $(r 6) = (6 - r 6)$ .
Method #2Process of Elimination
Step 1: Identify what is being asked
We need the row of Pascal's triangle corresponding to $n = 6$ , which must have exactly $6 + 1 = 7$ entries.
Step 2: Eliminate the row with 6 entries
$15101051$ has only 6 entries — this is row 5, used for $(a + b)^{5}$ , not $(a + b)^{6}$ .
Step 3: Eliminate the row with 8 entries
$172135352171$ has 8 entries — this is row 7, used for $(a + b)^{7}$ .
Step 4: Eliminate the incorrect 7-entry row
$161281261$ has 7 entries but the values are wrong. For example, $(2 6) = 15 \neq = 12$ . These appear to be powers of 2, not binomial coefficients.
Step 5: Select the correct answer
$1615201561$ is row 6 of Pascal's triangle, confirmed by $(2 6) = 15$ and $(3 6) = 20$ .

Free preview

Step-by-step walkthrough

Step 1: Write the formula

Step 2: Cancel common factors

Step 3: Compute numerator and denominator

Step 4: Divide to get the answer

Step 1: Identify what is being asked

Step 2: Eliminate 36

Step 3: Eliminate 84

Step 4: Eliminate 210

Step 5: Select the correct answer

Step-by-step walkthrough

Step 1: Identify the required row

Step 2: Compute the entries using $\binom{6}{r}$

Step 3: Match to the correct option

Step 1: Identify what is being asked

Step 2: Eliminate the row with 6 entries

Step 3: Eliminate the row with 8 entries

Step 4: Eliminate the incorrect 7-entry row

Step 5: Select the correct answer

15 more questions in this topic

Step-by-step walkthrough

Step 1: Write the formula

Step 2: Cancel common factors

Step 3: Compute numerator and denominator

Step 4: Divide to get the answer

Step 1: Identify what is being asked

Step 2: Eliminate 36

Step 3: Eliminate 84

Step 4: Eliminate 210

Step 5: Select the correct answer

Step-by-step walkthrough

Step 1: Identify the required row

Step 2: Compute the entries using $\binom{6}{r}$

Step 3: Match to the correct option

Step 1: Identify what is being asked

Step 2: Eliminate the row with 6 entries

Step 3: Eliminate the row with 8 entries

Step 4: Eliminate the incorrect 7-entry row

Step 5: Select the correct answer

15 more questions in this topic