1 = a sq no p[ 8/113] 2 =divisible by7 p [ 16/113]for both case like the divisible by 7 and sq no then p= 23/113

Problem: A box contains cards numbered 11 to 123. A card is drawn at random from the box. Find the probability that the number is a square number and a multiple of 7?

Solution:

To find the probability that the number drawn is both a square number and a multiple of 7, we need to determine the number of favorable outcomes and the total number of possible outcomes.

Favorable Outcomes:
To be a favorable outcome, the drawn number must meet two conditions:
1. The number must be a square number.
2. The number must be a multiple of 7.

Condition 1: The number must be a square number:
To determine the range of square numbers within the given set, we need to find the square root of the smallest and largest numbers.

- The smallest number in the given set is 11. The square root of 11 is approximately 3.316.
- The largest number in the given set is 123. The square root of 123 is approximately 11.090.

Hence, the square numbers within the range of 11 to 123 are 16, 25, 36, 49, 64, 81, 100, and 121.

Condition 2: The number must be a multiple of 7:
To determine the range of multiples of 7 within the given set, we need to find the smallest and largest multiples of 7 within the range 11 to 123.

- The smallest multiple of 7 in the given set is 14.
- The largest multiple of 7 in the given set is 119.

Hence, the multiples of 7 within the range of 11 to 123 are 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98, 105, 112, and 119.

Total Outcomes:
The total number of possible outcomes is the number of cards in the box, which is given as 123 - 11 + 1 = 113.

Calculating the Probability:
To find the probability, we divide the number of favorable outcomes by the total number of possible outcomes.

- The number of favorable outcomes is 8, as there are 8 numbers that are both square numbers and multiples of 7 within the given set.
- The total number of possible outcomes is 113.

Hence, the probability that the number drawn is a square number and a multiple of 7 is 8/113, which can be further simplified if required.

Therefore, the probability is 8/113.