Maheshwari wants to buy 7 floors in a 68 storied building numbered from 1 to 68. She follows these rules: (1) none of the floor numbers she can choose is a prime number, (2) no digit in the floor number is a prime digit (prime digits are 2, 3, 5 and 7), and (3) she cannot buy the 1st floor. How many different floor numbers satisfy all these rules and are therefore available for her to choose from?

Introduction / Context:
This question combines basic number theory ideas: prime numbers, prime digits and constraints on a range of building floor numbers. It tests careful filtering of numbers based on multiple conditions rather than complex calculation.

Given Data / Assumptions:

• The building has floors numbered from 1 to 68.
• Maheshwari cannot take the 1st floor.
• Any floor she can take must not itself be a prime number.
• None of the digits of an allowed floor number may be prime digits (2, 3, 5, 7).
• Prime digits are 2, 3, 5 and 7; non prime digits are 0, 1, 4, 6, 8 and 9.

Concept / Approach:
We must list all floor numbers between 2 and 68 that satisfy the two conditions: the entire number is not prime, and each digit is from the non prime digit set {0, 1, 4, 6, 8, 9}. This is a filtering problem. We systematically check each number, reject numbers with any digit 2, 3, 5 or 7, and then among the remaining ones, reject any prime numbers.

Step-by-Step Solution:
Step 1: Identify allowable digits: 0, 1, 4, 6, 8 and 9.Step 2: We cannot use floor 1 due to the given rule. So the candidate range is from 2 to 68.Step 3: List numbers from 2 to 68 whose digits are all in {0, 1, 4, 6, 8, 9}. These are: 4, 6, 8, 9, 10, 14, 16, 18, 40, 44, 46, 48, 49, 60, 64, 66 and 68.Step 4: Check which of these are prime. All are composite because they have obvious divisors (for example 4 = 2 * 2, 6 = 2 * 3, 8 = 2 * 4, 9 = 3 * 3, 10 = 2 * 5, 14 = 2 * 7, 16 = 4 * 4, 18 = 2 * 9, 40 = 4 * 10, 44 = 4 * 11, 46 = 2 * 23, 48 = 6 * 8, 49 = 7 * 7, 60 = 6 * 10, 64 = 8 * 8, 66 = 6 * 11, 68 = 4 * 17).Step 5: Since all these candidates are non prime and obey the digit rule, all of them are acceptable floors for Maheshwari.Step 6: Count them: there are 17 such floors.

Verification / Alternative check:
Cross check each option by attempting to count manually. There are no other two digit numbers in 1 to 68 with only allowed digits that also avoid primality, and none of the listed numbers violate the digit rule.Thus 17 is a consistent and verified count.

Why Other Options Are Wrong:
11 and 13 underestimate the count and would require discarding some valid floors without reason.19 overestimates the count, implying the existence of additional valid floors that either contain prime digits or are themselves prime.

Common Pitfalls:
Forgetting that digits 2, 3, 5 and 7 are forbidden even if the whole number is composite.Accidentally including floor 1, which is explicitly prohibited.Missing composite numbers like 49 or 46 due to not checking beyond small divisors. A systematic digit filter followed by a quick divisibility check avoids these errors.

Final Answer:
There are 17 possible floor numbers that Maheshwari can buy.