In a group of 120 secretaries, 82 of them are good at typing, 51 are good at both typing and shorthand, while 54 are poor at shorthand. Some secretaries are poor at both typing and shorthand.

a)How many are good at shorthand but poor at typing?
b)How many are poor at typing?

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To answer these questions, we can use a mathematical technique called the inclusion-exclusion principle. Let's break down the information given and solve each question step by step.

First, we know that there are 82 secretaries who are good at typing. This implies that there are at least 82 secretaries who are good at typing and shorthand, as being good at typing is a subset of being good at both typing and shorthand.

We also know that there are 51 secretaries who are good at both typing and shorthand. So, the number of secretaries who are good at shorthand but poor at typing can be found by subtracting the number of secretaries who are good at both from the total number of secretaries who are good at typing:

a) Number of secretaries good at shorthand but poor at typing = Number of secretaries good at typing - Number of secretaries good at both typing and shorthand
= 82 - 51
= 31

Therefore, there are 31 secretaries who are good at shorthand but poor at typing.

Next, we are given that 54 secretaries are poor at shorthand. This tells us nothing about their typing skills. However, we can use this information to find the number of secretaries who are poor at typing.

To do this, we need to calculate the total number of secretaries who are poor at typing, including those who are both poor at typing and shorthand.

b) Number of secretaries poor at typing = Total secretaries - (Number of secretaries who are good at typing - Number of secretaries good at both typing and shorthand + Number of secretaries poor at shorthand)
= 120 - (82 - 51 + 54)
= 120 - 82 + 51 - 54
= 135 - 136
= -1

Here, we see that the result is negative, which indicates that there might be an error in the given data. It is not possible to have a negative number of secretaries. Hence, we cannot determine the exact number of secretaries who are poor at typing with the given information.