Old McDonald's Chickens
Old McDonald has a farm. On this farm he has 5 white, 3 black and 2 brown chickens. They all have names, and are distinguishable. One day Old McDonald decides to sell two chickens of each color (6 total). In how many ways can he do it?
([5],[2])([3],[2])([2],[2]) = 30
Bob the Builder decided to build a rectangular shed, each side being one log long. He has plenty of wooden logs of five different lengths: 8 feet, 10 feet, 12 feet, 15 feet, and 20 feet, and he doesn’t want to cut them. How many different sizes can the shed be?
(NOTE: a rectangle 10’x15’ and a rectangle 15’x10’ are counted as the same size!)
([5],[2])+([5],[1])=15
N is a multiple of 72 that is greater than 7 million. What is the least possible value of N?
If x and y are positive integers such that x^2-y^2=84 , what is the greatest possible value of x^2 ?
Factor the difference of squares to find
(x-y)=2, (x+y)=42.
Then
x = 22 Rightarrow x^2=484.
How many two digit numbers are there in which the first digit is divisible by the second?
Add up the numbers of divisors of the tens digits to obtain 23.
Old McDonald has a farm. On this farm he has 5 white, 3 black and 2 brown chickens. Two of his chickens are named Pauline and Claudine. One day Old McDonald decides to make a video commercial and picks up two chickens randomly out of the 10 he has. What is the probability that both Pauline and Claudine were selected? [Recall that probability is the ratio of the number of selected outcomes to the number of possible outcomes.]
We want one pair out of ([10],[2]) .
Answer: 1/45
Four traveling friends decide to spend a night in a motel. The motel has four identical (indistinguishable) rooms where they may be placed for the night. If each room sleeps at most three people, how many different ways are there to place the friends into rooms?
Use 4 rooms: 1 solution.
Use 3 rooms: ([4] ,[2])=6
Use 2 rooms: ([4] ,[2]) div2+([4] ,[3]) = 7
Answer: 14
Consider the integers from 2 through 15. How many of them are divisors of 707,616?
Since 8 | 616, we have 2,4,8. Since the digit sum is 27, we have 3,9. Combining the previous results, we have 6,12. By the divisibility rule for 1001=7*11*13, we have 7, 13. Finally, since 14 = 7*2, we have 14.
That's 10 divisors in the given interval.
Find a divisor of 27,001 that lies strictly between 500 and 1000.
We can factor 27,001 as a sum of two cubes:
27,001=30^3+1^3
=(30+1)(30^2-1*30+1)=31(871).
The answer is 871.
Write down 7 consecutive integers so that the digit 2 is used exactly 16 times.
For example, 2215 through 2221.
Old McDonald has a farm. On this farm he has 5 white, 3 black and 2 brown chickens. How many selections of 5 chickens can he make, so that at least 3 of them are white?
Old McDonald could make a group with:
(a) exactly 3 white chickens --> ([5],[3])([5],[2])=100 ,
(b) exactly 4 --> ([5],[4]) times5=25 ,
(c) exactly 5 --> 1 possibility.
Answer: 126.
Seven justices of the Supreme Court are assembled for a photo session. There are four men and three women. In how many ways can they be seated in a row, so that at least two men sit next to each other?
7! - 4!times3! = 4896
Consider the 12 smallest positive even numbers. Their product is 19619905J36K0 , where two of the digits, marked by J and K, have been concealed.
What is the value of J-K?
The product
2times4times...times22times24
has 10 and 20 as factors. Thus it ends in two zeros, so K = 0. It also has 22 as a factor, so it is divisible by 11. By the divisibility rule for 11, J = 5. Thus
J-K=5.
Find two positive integers whose difference is less than 100 and whose product is
2111^2 - 175(2111) + 87(88)
Let 2111 = x. Then we have
x^2-(87+88)x +87(88)
=(x-87)(x-88)
So the two factors we want are 2111 - 87 and 2111 - 88, i.e. 2023 and 2024.
Using two straight lines, divide the face of a standard 12-hour clock into three parts so that the sum of numbers in each of three parts is the same as in any other part.
Answer:
{11,12,1,2}, {10,9,3,4}, {5,6,7,8}