Combinatorics practice problems - q5 pending

1. A telephone number has the form ABC – DEF – GHIJ, where each letter represents a different digit. The digits in each part of the numbers are in decreasing order; that is, A > B > C, D > E > F, and G > H > I > J. Furthermore, D, E, and F are consecutive even digits; G, H, I, and J are consecutive odd digits; and A + B + C = 9. Find A.
Answer: 8

Solution:
GHIJ can be 9,7,5,3 or 7,5,3,1.
If it's the later, A+B+C > 9 so it's the former.
Now try with DEF = 864
A+B+C = 2+1+0 wrong
Now try with DEF = 642
A+B+C = 8+1+0 correct, so A = 8.

2. Define na! for n and a positive to be
n_a! = n(n – a) (n – 2a) (n – 3a) ..... (n – ka), where k is the greatest integer for which n > ka. Then the quotient

72_8!/18_2! is equal to 2^x. Find x.
Answer: 18

Solution: quite simple.

3. How many ways can a student schedule 3 mathematics courses – algebra, geometry, and number theory in a 6-period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other 3 periods is of no concern here).
Answer: 24

Solution:
Let the other courses be A,B,C.
Then there are 4 empty slots around them _A_B_C_.
Choose 3 out of them and permute : 4C3.3! = 24.

4. There are 20 students participating in an after-school program offering classes in yoga, bridge, and painting. Each student must take at least one of these three classes, but may take two or all three. There are 10 students taking yoga, 13 taking bridge, and 9 taking painting. There are 9 students taking at least two classes. How many students are taking all three classes?
Answer: 3

Solution:
It's clear that 11 students are taking 1 class each.
Rest 9, let's say they are taking 2 classes each.
Then there is shortfall of 3(10+13+9 - 11 - 18 = 3).
It means 3 students are taking 3 classes each.

5. Five friends sat in a movie theater in a row containing 5 seats, numbered 1 to 5 from left to right. (The directions ”left” and ”right” are from the point of view of the people as they sit in the seats.) During the movie Asha went to the lobby to get some popcorn. When she returned, she found that Bhawna had moved two seats to the right, Chetna had moved one seat to the left, and Divya and Esha had switched seats, leaving an end seat for Asha. In which seat had Asha been sitting before she got up?

Answer: 2

Solution:
By trial and error.
Seating before : BACDE(1,2,3,4,5)
Seating after:     ACBED(1,2,3,4,5)

So D,E exchanged seats.
B shifted right by 2.
C shifted left by 1.

Notes for original seating arrangement:
1. D,E both can't be sitting on edge seats
2. B can be on 1,2,3
3. C can be on 2,3,4,5

6. A palindrome number is the same either read from left to right or right to left, for example. 121 is a palindrome number. Total possible 5-digit palindrome numbers can be written as 10a . Find the value of a.
Answer: 90

Solution:
First digit can be chosen in 9 ways, second in 10 ways. 4th and 5th in 1 way. Third in 10 ways.

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