Question

The number of \(3\)'s that are preceded by \(5\) but not followed by \(2\) in the following sequence of digits is \(3147531245321887538162537531675324\).

• A. \(7\)
• B. \(5\)
• C. \(4\)
• D. \(6\)

Hint:

For sequence-counting questions, check both the previous and next digit carefully before counting.

Answer 1

The Correct Option is C

Concept:
We have to count only those \(3\)'s which satisfy both conditions: \[ \text{Previous digit must be }5 \] and \[ \text{Next digit must not be }2 \] So, we need to look for the pattern: \[ 5\,3\,x \] where: \[ x\neq 2 \]

Step 1: Write the given digit sequence.
The given sequence is: \[ 3147531245321887538162537531675324 \]

Step 2: Find all \(3\)'s preceded by \(5\).
Now check every occurrence of \(3\) in the sequence and see whether the digit before it is \(5\). The useful parts are: \[ 753 \] Here, \(3\) is preceded by \(5\). \[ 532 \] Here also, \(3\) is preceded by \(5\), but it is followed by \(2\). \[ 753 \] Here, \(3\) is preceded by \(5\). \[ 253 \] Here, \(3\) is preceded by \(5\). \[ 753 \] Here, \(3\) is preceded by \(5\). \[ 532 \] Here, \(3\) is preceded by \(5\), but it is followed by \(2\).

Step 3: Remove the cases where \(3\) is followed by \(2\).
The condition says \(3\) should not be followed by \(2\). So, cases like: \[ 532 \] are not counted. There are two such rejected cases: \[ 532,\quad 532 \]

Step 4: Count the valid cases.
The valid cases are: \[ 753,\quad 753,\quad 253,\quad 753 \] In each of these, the digit \(3\) is preceded by \(5\), and the digit after \(3\) is not \(2\). So, total valid \(3\)'s are: \[ 4 \]

Step 5: Final answer.
Therefore, the required number of \(3\)'s is: \[ 4 \] \[ \therefore \text{Correct Answer is (C)} \]