The number of solution of the in equation `""^10C_(x-1) gt 3. ""^10C_x ` is :

Options

A. 0
B. 1
C. 2
D. 9

Explanation

To solve the inequality \( \binom{10}{x-1} > 3 \cdot \binom{10}{x} \), we will follow these steps:

### Step 1: Write the binomial coefficients in terms of factorials
The binomial coefficient \( \binom{n}{r} \) is defined as:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
Thus, we can express \( \binom{10}{x-1} \) and \( \binom{10}{x} \) as:
\[ \binom{10}{x-1} = \frac{10!}{(x-1)!(10-(x-1))!} = \frac{10!}{(x-1)!(11-x)!} \]
\[ \binom{10}{x} = \frac{10!}{x!(10-x)!} \]

### Step 2: Substitute the binomial coefficients into the inequality
Substituting these into the inequality gives:
\[ \frac{10!}{(x-1)!(11-x)!} > 3 \cdot \frac{10!}{x!(10-x)!} \]

### Step 3: Cancel \( 10! \) from both sides
Since \( 10! \) is common on both sides, we can cancel it:
\[ \frac{1}{(x-1)!(11-x)!} > \frac{3}{x!(10-x)!} \]

### Step 4: Rearrange the inequality
Rearranging gives:
\[ \frac{x!(10-x)!}{(x-1)!(11-x)!} > 3 \]
Notice that \( x! = x \cdot (x-1)! \) and \( (11-x)! = (11-x)(10-x)! \). Substituting these in gives:
\[ \frac{x \cdot (10-x)!}{(11-x)(10-x)!} > 3 \]
This simplifies to:
\[ \frac{x}{11-x} > 3 \]

### Step 5: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ x > 3(11 - x) \]
Expanding this results in:
\[ x > 33 - 3x \]

### Step 6: Combine like terms
Adding \( 3x \) to both sides results in:
\[ 4x > 33 \]
Dividing by 4 gives:
\[ x > \frac{33}{4} = 8.25 \]

### Step 7: Determine integer solutions
Since \( x \) must be an integer, we have:
\[ x \geq 9 \]
However, since \( x \) must also satisfy the condition of the binomial coefficient, \( x \) can be at most 10 (since \( \binom{10}{x} \) is defined for \( x \leq 10 \)).

### Step 8: List possible integer solutions
Thus, the possible integer values for \( x \) are:
- \( x = 9 \)
- \( x = 10 \)

### Conclusion
Therefore, there are **2 solutions** to the inequality \( \binom{10}{x-1} > 3 \cdot \binom{10}{x} \).

### Final Answer
The number of solutions is \( 2 \).

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