CALCULATOR, FORMULAS, TECHNIQUES & SHORT-CUTS FOR MATHEMATICS 1. A committee of 3 members is to be formed consisting of one representative each from labor, management, and the public. If there are 3 possible representatives from labor, 2 from management, and 4 from the public, determine how many different committees can be formed. A. 16
B. 24 *
A. 8,709,120 *
Formula: Fundamental Principle of counting:mxn ways 2. In how many ways can 5 differently colored marbles be arranged in a row? A. 120 *
B. 140
3. In how many ways can 10 people be seated on a bench if only 4 seats are available?
Solution:
C. 4050
A. 2520 *
P=
nPr 4. It is required to seat 5 men and 4 women in a row so that the women occupy the even places. How many such arrangements are possible? B. 8502
Solution: m x n ways
m = 5! → To seat 5 men in Odd places n = 4! → To seat 4 women in Even places 5. How many 4-digit numbers can be formed with the 10 digits 0, 1, 2, 3 . . . 9 if the last digit must be zero and repetitions are not allowed? A. 4536
B. 504 *
Use counting principle: m x n ways. Start with the most critical digit, the last digit. Then, the first digit.
RBF
B. 5022
n! q!r !s!...
8. In how many ways can 7 people be seated at a round table if 2 particular people must not sit next to each other? A. 460
B. 480 *
Solve the total ways without restriction: PT = ( n − 1) ! → Circular Permutation Solve the total ways as 2 particular people sit next to each other. Then subtract it to the total. 9. In how many ways can 10 objects be split into two groups containing 4 and 6 objects, respectively? A. 110
B. 210 *
nCr
11. Out of 5 mathematicians and 7 physicists, a committee consisting of 2 mathematicians and 3 physicists is to be formed. In how many ways can this be done if one particular physicist must be on the committee? B. 122
m x n ways Since this is grouping, use combination, For physicist, only 2 are need from 6. 12. Out of 5 mathematicians and 7 physicists, a committee consisting of 2 mathematicians and 3 physicists is to be formed. In how many ways can this be done if two particular mathematicians cannot be on the committee? A. 150
B. 105 *
m x n ways Since this is grouping, use combination, For mathematians, only 3 are qualified for committee. 13. How many different salads can be made from lettuce, escarole, endive, watercress, and chicory? A. 30 B. 31 *
2n − 1
B. 250
A.
9 * 13
C.
10 13
PA = 1 − PNot A PE or F = PE + PF − PE and F → with common outcomes 18. A ball is drawn at random from a box containing 6 red balls, 4 white balls, and 5 blue balls. Determine the probability that it is red or white. 2 5 A. * B. 3 7
PE or F = PE + P F 19. A fair die is tossed twice. Find the probability of getting a 4, 5, or 6 on the first toss and a 1, 2, 3, or 4 on the second toss. A.
14. From 7 consonants and 5 vowels, how many words can be formed consisting of 4 different consonants and 3 different vowels? The words need not have meaning.
2 5
B.
1 * 3
For independent events: PA x PB 20. Find the probability of a 4 turning up at least once in two tosses of a fair die.
Answer: 1,764,000
m x n ways This is combination of grouping and arrangement.
A. 6 *
N −1 ; if N ( sum ) ≤ 6 36 13 − N P= ;if N ( sum ) ≥ 7 36 1 Answer: 9 17. A card is drawn at random from an ordinary deck of 52 playing cards. Find the probability that it is neither a four nor a club. P=
15. For what value of n is 3 i n +1C3 = 7 i nC2 .
10. In how many ways can a committee of 5 people be chosen out of 9 people? A. 126 *
nCr
A. 150 *
Consider the mathematics books as a group and assume as 1 book. The math books are arrange in a group.
Formula:
A. 2880 *
C. 709,812
7. Five red marbles, two white marbles, and three blue marbles are arranged in a row. If all the marbles of the same color are not distinguishable from each other, how many different arrangements are possible?
Formula: nPn = n!
A. 5040 *
6. Four different mathematics books, six different physics books, and two different chemistry books are to be arranged on a shelf. How many different arrangements are possible if only the mathematics books must stand together?
B. 8
16. Two dice are rolled; find the probability of getting a sum of 9.
A.
1 6
B.
11 * 36
Re peated Trials Probability: P = C(n ,r) pr qn − r 21. One bag contains 4 white balls and 2 black balls; another contains 3 white balls and 5 black balls. If
CALCULATOR, FORMULAS, TECHNIQUES & SHORT-CUTS FOR MATHEMATICS one ball is drawn from each bag, find the probability that one is white and one is black. A.
13 * 24
B.
5 12
Expectation = ∑ ( prize )( probability ) Ticket price is the equal to expected amount of money to be won. Answer: P0.20
Independent events and the situation can be interchange.
25. Find the mean of 8, 15, 19, 24, and 14. A. 12
22. The probability that a regularly scheduled flight departs on time is P(D) = 0.83, the probability that it arrives on time is P(A) = 0.92, and the probability that it departs and arrives on time is 0.78. Find the probability that a plane (a) arrives on time given that it departed on time, and (b) departed on time given that it has arrived on time. a. 0.94 b. 0.85
P (B / A ) =
26. Find the median for 8, 15, 19, 24, and 14. A. 15 *
B. 24
27. Find the mode of 24, 17, 32, 17, 15, and 19.
P ( A and B )
A. 17 *
B. 15
P(A)
Mode is the value that occur most frequent.
where : P (B / A ) = Pr obability of B given event A happen. P ( A and B ) = Pr obability of A and B happen P ( A ) = Pr obability of event A
23. The probability that both stages of a 2-stage missile will function correctly is 0.95. The probability that the first stage will function correctly is 0.98. What is the probability that the second stage will function correctly given that the first one does? Answer: 0.97
Conditional Probability: P ( A and B ) P(A)
24. In a lottery there are 200 prizes of P5, 20 prizes of P25, and 5 prizes of P100. Assuming that 10,000 tickets are to be issued and sold, what is a fair price to pay for a ticket?
RBF
x1 + x 2 + ⋯ + x n n
Arrange from lowest to highest and find the middle.
Conditional Probability :
P (B / A ) =
x=
B. 16 *
28. Find the range for 8, 15, 19, 24, and 14. A. 24
C. 16 *
Range:Highest − Lowest 29. Find the standard deviation: 10, 12, 15, 18, and 20. Answer: 3.68
Use calcu.
