Discrete Math Explain, without using a truth table, why (p or q or r) and ( p or

Question

Discrete Math

Explain, without using a truth table, why (p or q or r) and ( p or q or r) is true when at least one of p, q, and r is true and at least one is false, but is false when all three variables have the same truth value. Seven teachers and three children are to stand in line for a photo session. a) How many arrangements are there? b) How many arrangements are there if only any 4 teachers and no children are in the picture? Twenty documents must be moved from a computer's desktop into folders A or B. The order in which these documents are moved is irrelevant, and no document must be in both folders. a) Find the number of ways this can be done. b) Find the number of ways this can be done if no folder can remain empty.

Explanation / Answer

Question 7

7 teachers and 3 children

total 10

a) we need to arrange 10 at 10 places

total number of arangement = 10! = 3628800

b)

If only 4 teachers in photo

First choose 4 teachers out of 7 = 7C4

then arrange 4 teachers at 4 places = 4!

Total number of arrangments = 7C4 * 4! = 7P4 = 7! / 3! = 840

Question 8

a)

Number of ways to move 1st document = 2 ( A or B)

Number of ways to move 2nd document = 2 ( A or B)

Number of ways to move 3rd document = 2 ( A or B)

and so on

Number of ways to move 20th document = 2 ( A or B)

Total number of ways = 2*2*2* ...... upto 20 terms = 220 = 1048576

b)

Number of ways when all document in same folder = 2 (Either all in A or all in B)

total number of ways = 1048576

Number of ways when no folder is empty = 1048576 - 2 = 1048574

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