id null | metadata null | instruction stringlengths 25 140 | input stringclasses 1
value | output stringlengths 1.22k 2.52k |
|---|---|---|---|---|
null | null | What is the coefficient of $x^2y^6$ in the expansion of $\left(\frac{3}{5}x-\frac{y}{2}\right)^8$? Express your answer as a common fraction | To find the coefficient of \( x^2y^6 \) in the expansion of \( \left(\frac{3}{5}x - \frac{y}{2}\right)^8 \), we will use the Binomial Theorem, which states that:
\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\]
In our case, we can identify \( a = \frac{3}{5}x \) and \( b = -\frac{y}{2} \), and \( n = 8 \).
... | |
null | null | how many r in strawberry? | To determine how many times the letter "r" appears in the word "strawberry," we will follow a systematic approach.
### Step 1: Analyze the problem requirements
The problem requires us to count the occurrences of the letter "r" in the word "strawberry." This is a straightforward counting problem where we need to identi... | |
null | null | how many e in 'The quick brown fox jumps over the lazy dog.'? | To determine how many times the letter "e" appears in the sentence "The quick brown fox jumps over the lazy dog," we will follow a systematic approach.
### Step 1: Analyze the problem requirements
The problem requires us to count the occurrences of the letter "e" in the given sentence. This is a straightforward counti... | |
null | null | how many t in 'She sells seashells by the seashore.'? | To determine how many times the letter "t" appears in the sentence "She sells seashells by the seashore," we will follow a systematic approach.
### Step 1: Analyze the problem requirements
The problem requires us to count the occurrences of the letter "t" in the given sentence. This is a straightforward counting probl... | |
null | null | how many a in 'Aardvarks are amazing animals.'? | To determine how many times the letter "a" appears in the sentence "Aardvarks are amazing animals," we will follow a systematic approach.
### Step 1: Analyze the problem requirements
The problem requires us to count the occurrences of the letter "a" in the given sentence. This includes both uppercase "A" and lowercase... | |
null | null | how many o in 'Once upon a time in a land far, far away.'? | To determine how many times the letter "o" appears in the sentence "Once upon a time in a land far, far away," we will follow a systematic approach.
### Step 1: Analyze the problem requirements
The problem requires us to count the occurrences of the letter "o" in the given sentence. This includes both uppercase "O" an... | |
null | null | how many i in 'The significance of this discovery is immense.'? | To determine how many times the letter "i" appears in the sentence "The significance of this discovery is immense," we will follow a systematic approach.
### Step 1: Analyze the problem requirements
The problem requires us to count the occurrences of the letter "i" in the given sentence. This includes both uppercase "... | |
null | null | how many s in 'Success is not final, failure is not fatal.'? | To determine how many times the letter "s" appears in the sentence "Success is not final, failure is not fatal," we will follow a systematic approach.
### Step 1: Analyze the problem requirements
The problem requires us to count the occurrences of the letter "s" in the given sentence. This includes both uppercase "S" ... | |
null | null | how many r in 'The rain in Spain falls mainly on the plain.'? | To determine how many times the letter "r" appears in the sentence "The rain in Spain falls mainly on the plain," we will follow a systematic approach.
### Step 1: Analyze the problem requirements
The problem requires us to count the occurrences of the letter "r" in the given sentence. This includes both uppercase "R"... | |
null | null | how many n in 'Necessity is the mother of invention.'? | To determine how many times the letter "n" appears in the sentence "Necessity is the mother of invention," we will follow a systematic approach.
### Step 1: Analyze the problem requirements
The problem requires us to count the occurrences of the letter "n" in the given sentence. This includes both uppercase "N" and lo... | |
null | null | how many l in 'Lollipops are loved by little children.'? | To determine how many times the letter "l" appears in the sentence "Lollipops are loved by little children," we will follow a systematic approach.
### Step 1: Analyze the problem requirements
The problem requires us to count the occurrences of the letter "l" in the given sentence. This includes both uppercase "L" and ... | |
null | null | how many c in 'Circumstances can change quickly.'? | To determine how many times the letter "c" appears in the sentence "Circumstances can change quickly," we will follow a systematic approach.
### Step 1: Analyze the problem requirements
The problem requires us to count the occurrences of the letter "c" in the given sentence. This includes both uppercase "C" and lowerc... |
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