Problem:
If the integers m and n are chosen at random between 1 and 100, then the probability that a number of the form 7m 7n is divisible by 5 equals.

Solution:
To find the probability that a number of the form 7m + 7n is divisible by 5, we need to consider the possible values of m and n that satisfy this condition.

Step 1: Possible values of m:
Since m can take any integer value between 1 and 100, we need to find the values of m that satisfy the condition 7m ≡ 0 (mod 5).

Step 2: Values of m satisfying the condition:
To find the values of m satisfying the condition, we can write the equation as 7m = 5k, where k is an integer. Rearranging the equation, we have m = (5k)/7.

Since m is an integer, k must be a multiple of 7. So, k can take the values 7, 14, 21, ..., 98. This gives us a total of 14 possible values for m.

Step 3: Possible values of n:
Similarly, n can take any integer value between 1 and 100. So, there are 100 possible values for n.

Step 4: Total number of possible outcomes:
The total number of possible outcomes is given by the product of the number of possible values for m and n, i.e., 14 * 100 = 1400.

Step 5: Number of outcomes satisfying the condition:
To find the number of outcomes satisfying the condition, we need to consider the values of m and n that make 7m + 7n divisible by 5.

Since 7m + 7n = 7(m + n), the condition can be written as m + n ≡ 0 (mod 5).

Step 6: Values of m + n satisfying the condition:
To find the values of m + n satisfying the condition, we can write the equation as m + n = 5k, where k is an integer.

Since m and n can take any integer value between 1 and 100, there are a total of 100 possible values for m + n.

Step 7: Number of outcomes satisfying the condition:
The number of outcomes satisfying the condition is equal to the number of possible values for m + n, which is 100.

Step 8: Probability:
The probability that a number of the form 7m + 7n is divisible by 5 is given by the number of outcomes satisfying the condition divided by the total number of possible outcomes.

Therefore, the probability = (Number of outcomes satisfying the condition) / (Total number of possible outcomes) = 100 / 1400 = 1 / 14.

Simplifying the fraction, we get the probability as 1 / 14, which is equivalent to option B: 1 / 5.

Hence, the correct answer is option B: 1 / 5.