In the expansion of `(6)^(10)xx(7)^(12)xx(5)^(55)xx(11)^(121)`, numbers of prime numbers are?

Options

A. 213
B. 222
C. 211
D. 214

Answer

To find the number of prime numbers in the expansion of \( (6)^{10} \times (7)^{12} \times (5)^{55} \times (11)^{121} \), we can follow these steps:

### Step 1: Factor the base numbers into prime factors
- The number \( 6 \) can be factored into primes as \( 6 = 2 \times 3 \).
- Therefore, \( (6)^{10} = (2 \times 3)^{10} = 2^{10} \times 3^{10} \).

### Step 2: Write the expression with prime factors
- Now we can rewrite the entire expression:
\[ (6)^{10} \times (7)^{12} \times (5)^{55} \times (11)^{121} = (2^{10} \times 3^{10}) \times (7^{12}) \times (5^{55}) \times (11^{121}) \]
- This simplifies to:
\[ 2^{10} \times 3^{10} \times 5^{55} \times 7^{12} \times 11^{121} \]

### Step 3: Identify the prime numbers
- The prime numbers in the expression are \( 2, 3, 5, 7, \) and \( 11 \).

### Step 4: Count the unique prime numbers
- The unique prime numbers we identified are:
- \( 2 \)
- \( 3 \)
- \( 5 \)
- \( 7 \)
- \( 11 \)

### Step 5: Total count of prime numbers
- There are a total of **5 unique prime numbers** in the expression.

### Final Answer
The number of prime numbers in the expansion is **5**.

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