Problem Solving - 56

A certain stock exchange designates each stock with a one- , two-,or three-letter code ,where each letter is selected from the 26 letters of the alphabet. If the letter may be repeated and if the same letters used in a different order constitute a different code, how many different stocks is it possible to uniquely designate with these codes?

A) 2951

B) 8125

C) 15600

D) 15302

E) 18278

Answer: E

Total ways to pick a 3 letter code = 26*26*26 = 26^3
Total ways to pick a 2 letter code = 26*26 = 26^2
Total ways to pick a 1 letter code = 26

Thus total stocks: 26^3 + 26^2 + 26 = 26(26^2 + 26 + 1) = 18278

Problem Solving - 46

A certain restaurant offers 6 kinds of cheese and 2 kinds of fruit for its dessert platter. If each dessert platter contains an equal number of kinds of cheese and kinds of fruit, how many different dessert platters could the restaurant offer?

A) 8
B) 12
C) 15
D) 21
E) 27

Answer: E

Kinds of platter:

1 cheese + 1 fruit
Total = 6 * 2 = 12 types of platters

2 cheese + 2 fruit
Total = 6C2 * 2C2 = 15 * 1 = 15 types

Total: 12 + 15 = 27

Problem Solving - 44

Samar tried to type his new 7-digit phone number on a form, but what appeared on the form was 39269, since the '4' key on his computer no longer works. His secretary has decided to make a list of all of the numbers that could be Samar's new number. How many numbers will there be on the list?

(A) 21
(B) 24
(C) 25
(D) 30
(E) 36

Answer: A

It is clear from the question that there are two 4's missing as it is a seven digit number.
Total number of ways of choosing places for these two missing 4's in these 7 digits is 7C2 = 21 Once you fix the place for these two 4's rest all numbers will occupy remaining places.

Problem Solving - 22

A certain restaurant offers 6 kinds of cheese and 2 kinds of fruit for it dessert platter. If each dessert platter contains an equal number of kinds of cheese and kinds of fruit, how many different dessert platters could the restaurant offer?

(A) 8

(B) 12

(C) 15

(D) 21

(E) 27

Answer: E

There are two kinds of platter ---- 1 cheese + 1 fruit

Total types of platter -- 6 * 2= 12 ---(1)

For 2 cheese + 2 fruits

Total -- 6C2 * 2C2 = 15 * 1 = 15 ---(2)

Hence adding (1) and (2) we get total = 12 + 15 = 27

Problem Solving - 21

Mrs. Napueta has been given film vouchers. Each voucher allows the holder to see a film without charge. She decides to distribute them among her four nephews so that each nephew gets at least two vouchers. How many vouchers has Mrs. Napueta been given if there are 120 ways that she could distribute the vouchers?

(A) 13

(B) 14

(C) 15

(D) 16

(E) more than 16


Answer: C

Formula - Number of ways of distributing n identical things among r persons when each person may get any number of things = C(n+r-1, r-1)

Hence

Let total number of vouchers = n

All 4 nephews get atleast 2 vouchers, thus remaining vouchers to be distributed among 4 nephews where each can get 0 or more vouchers = n-8

Using the formula above we get = C(n-8+4-1, 3) = (n-8+4-1)C3 = 120 = (n-5)(n-6)(n-7) = 120 * 3! = 720

Hence n = 15