1.How many positive four-digit integers can be formed that are even?9x9x9x5=3645
2.Three differently colored six-sided dice are rolled. In how many different ways can the dice turn up?
6x6x6=216
3.How many positive integers five-digit integers start with the digit 1?1x9x9x9x9=6561
4.How many positive four-digit integers can be formed if there are no restrictions? 9999x2=19998
The ones I don't know
5.How many signals can be made by raising three flags, one above the other, if there are 6 differently colored flags from which to choose?
6.In how many ways can all of the letters of the word HEXAGON be arranged if the consonants and vowels must alternate in their positions in the word?
7.The S.R.C. decided to run a "Guess the Baby Picture" contest. Six members brought their baby pictures to school and made a poster placing their names along the left side and their baby pictures, in random order, along the right side. Students entering the contest were to match the name with the correct baby picture. If each entry costs $1.00, how much would a student have to spend in order to guarantee a win?
#1 what about zero? 9*10*10*5
#2 ok
#3 as in #1, 1*10^4
#4 1000..9999 = ?
#5 6P3
#6 4! * 3!
#7 6 (assuming a win means matching a picture to the chosen name)
For number 4 will it be 8999 but times it by two since its asking integers so negative numbers count?
5. To find the number of different ways to raise three flags, one above the other, if there are 6 differently colored flags to choose from, we use the concept of combinations.
In this case, we want to select 3 flags out of 6 available flags.
The formula for combinations is given by:
C(n, r) = n! / (r!(n - r)!),
where n is the total number of objects and r is the number of objects we want to select.
Plugging in the values, we have:
C(6, 3) = 6! / (3!(6 - 3)!)
= 6! / (3! * 3!)
Simplifying further:
C(6, 3) = (6 * 5 * 4 * 3! * 2 * 1) / (3! * 3! * 2 * 1)
= (6 * 5 * 4) / (3! * 2)
Evaluating the factorials:
C(6, 3) = (6 * 5 * 4) / (3 * 2)
= 120 / 6
= 20
Therefore, there are 20 different ways to raise three flags, one above the other, if there are 6 differently colored flags to choose from.
6. To find the number of ways to arrange the letters of the word "HEXAGON" such that the consonants and vowels alternate in their positions, we can treat the consonants (H, X, G, N) and vowels (E, A, O) as two separate groups.
The consonants can be arranged in 4! (4 factorial) ways, and the vowels can be arranged in 3! (3 factorial) ways.
Since the two groups should alternate, the total number of arrangements would be the product of these two values:
Total arrangements = 4! * 3!
Calculating the factorials:
Total arrangements = 4 * 3 * 2 * 1 * 3 * 2 * 1
= 144
Therefore, there are 144 different ways to arrange the letters of the word "HEXAGON" such that the consonants and vowels alternate in their positions.
7. To guarantee a win in the "Guess the Baby Picture" contest, the student would need to match all 6 names correctly with the baby pictures.
For each name, there are 6 possible baby pictures to match with, and the student must spend $1.00 for each entry.
Therefore, the student would need to spend 6 entries (one for each name) multiplied by $1.00 per entry, which gives a total of $6.00 to guarantee a win.