+11 Hello,
To form an odd 4-digit number using 1, 2, 3, and 4, the units digit must be 1 or 3.
Case 1: The units digit is 1. We have 3 choices for the thousands digit (2, 3, or 4) and 2 choices for the hundreds digit (the remaining digits after the thousands digit and the units digit have been chosen). After the thousands and hundreds digits have been chosen, there is only 1 choice for the tens digit. Therefore, there are 3×2×1 = 6 ways to form an odd 4-digit number with 1 as the units digit.
Case 2: The units digit is 3. We have 3 choices for the thousands digit (2, 3, or 4) and 2 choices for the hundreds digit. After the thousands and hundreds digits have been chosen, there is only 1 choice for the tens digit. Therefore, there are 3×2×1 = 6 ways to form an odd 4-digit number with 3 as the units digit.
In total, there are 6+6=12 distinct odd 4-digit numbers that can be formed using the digits 1, 2, 3, and 4 if no digit may be used more than once. Nexusiceland.co.uk
+11 Hello,
To find the number of cozy 2-digit numbers, we can list all possible cozy numbers that satisfy the condition:
22, 20, 24, 26, 28, 02, 04, 06, 08
Therefore, there are 9 cozy 2-digit numbers.
To find the number of cozy 3-digit numbers, we can use a similar approach. Since the first and last digits of the 3-digit number can be any even digit (0, 2, 4, 6, 8), we only need to consider the middle digit to determine if the number is cozy. There are two cases to consider:
Case 1: The middle digit is even
In this case, we have 5 choices for the first digit (0, 2, 4, 6, 8), and 5 choices for the last digit (0, 2, 4, 6, 8). Therefore, there are 5 x 5 = 25 cozy 3-digit numbers that have an even middle digit.
Case 2: The middle digit is odd
In this case, the middle digit must be next to an even digit, so it must be either 1, 3, 5, or 7. We have 5 choices for the first and last digit (0, 2, 4, 6, 8), and 4 choices for the middle digit. Therefore, there are 5 x 4 x 5 = 100 cozy 3-digit numbers that have an odd middle digit.
Thus, the total number of cozy 3-digit numbers is 25 + 100 = 125.
Therefore, there are 9 cozy 2-digit numbers and 125 cozy 3-digit numbers. you can see also on tutorial CheckMyRota Login
+11 Hello,
To find the coefficient of u^2 v^9 in the expansion of (2u - 3v + u^2 - v^2)^9, we need to use the binomial theorem to expand the expression and then identify the term that contains u^2 v^9.
The binomial theorem states that the expansion of (a + b)^n can be written as:
(a + b)^n = ∑(k=0 to n) [n choose k] a^(n-k) b^k
where [n choose k] is the binomial coefficient, given by:
[n choose k] = n! / (k! (n-k)!)
Using this formula, we can expand (2u - 3v + u^2 - v^2)^9 as:
(2u - 3v + u^2 - v^2)^9 = ∑(k=0 to 9) [9 choose k] (2u)^(9-k) (-3v)^k (u^2)^{9-k} (-v^2)^k
= ∑(k=0 to 9) [9 choose k] 2^(9-k) (-3)^k u^(18-2k) v^k u^(2k) (-1)^k v^(2k)
= ∑(k=0 to 9) [9 choose k] 2^(9-k) (-3)^k (-1)^k u^18 v^k
= [9 choose 9] 2^0 (-3)^9 (-1)^9 u^18 v^9 + [9 choose 7] 2^2 (-3)^7 (-1)^7 u^14 v^7 + ...
We are interested in the coefficient of u^2 v^9, which appears only in the first term of the expansion. Therefore, the coefficient is given by:
[9 choose 9] 2^0 (-3)^9 (-1)^9 = 1 * (-19683) * (-1) = 19683
Therefore, the coefficient of u^2 v^9 in the expansion of (2u - 3v + u^2 - v^2)^9 is 19683. You can also visit Check My Rota Login
+11 Hello gm dear,
very nice question , well now we will see how it can be solved
There are two cases to consider: one where the two same-colored beads are next to each other, and one where they are not.
Case 1: Two same-colored beads are next to each other
In this case, we can treat the two same-colored beads as a single "unit". We then have five units to arrange: the "unit" of the two same-colored beads and the four units of the other color. The number of ways to arrange five units is 5! = 120. However, we must divide by 2! to correct for overcounting the arrangements of the two same-colored beads within the "unit". Therefore, there are 120/2! = 60 arrangements in this case.
Case 2: Two same-colored beads are not next to each other
In this case, we can treat the two same-colored beads as two separate units. We then have six units to arrange: the two units of the same-colored beads and the four units of the other color. The number of ways to arrange six units is 6! = 720. However, we must divide by 2! to correct for overcounting the arrangements of the two same-colored beads within their units, and we must also divide by 2 to correct for overcounting the arrangements that are reflections of each other. Therefore, there are 720/(2! * 2) = 180 arrangements in this case.
Overall, the total number of different ways that Joanna can assemble her bracelet is the sum of the two cases, which is 60 + 180 = 240.
hope so it will helps or you can see at BenefitsCal App
+11 Hello
I have a suggestion for you let see
(a) To find the equations of the medians, we first need to find the coordinates of the midpoints D, E, and F:
D: midpoint of BC = ((9+4)/2, (2+1)/2) = (6.5, 1.5)
E: midpoint of AC = ((-5+4)/2, (4+1)/2) = (-0.5, 2.5)
F: midpoint of AB = ((-5+9)/2, (4+2)/2) = (2, 3)
Now we can find the equations of the medians using the point-slope form:
AD: passes through A(-5,4) and D(6.5,1.5)
Slope of AD = (1.5-4)/(6.5+5) = -0.25
Equation of AD: y - 4 = -0.25(x + 5) or y = -0.25x + 5.25
BE: passes through B(9,2) and E(-0.5,2.5)
Slope of BE = (2.5-2)/(-0.5-9) = 0.05
Equation of BE: y - 2 = 0.05(x - 9) or y = 0.05x - 0.55
CF: passes through C(4,1) and F(2,3)
Slope of CF = (3-1)/(2-4) = -1
Equation of CF: y - 1 = -1(x - 4) or y = -x + 5
(b) To show that the three medians pass through the same point, we can find the point of intersection of any two medians, and then check that the third median also passes through that point.
Let's find the point of intersection of medians AD and BE. Setting the equations equal, we have:
-0.25x + 5.25 = 0.05x - 0.55
0.3x = 5.8
x ≈ 19.33
Substituting x into either equation gives us the corresponding y-coordinate:
y = -0.25(19.33) + 5.25 ≈ 0.92
So the point of intersection of medians AD and BE is approximately (19.33, 0.92).
Now we need to check if median CF also passes through this point. Substituting x = 19.33 into the equation of CF, we get:
y = -19.33 + 5 = -14.33
So the point (19.33, 0.92) does not lie on median CF. This may be due to rounding errors, but in general, the medians of a triangle do not necessarily intersect at a single point unless the triangle is equilateral. However, they do always intersect at a common point called the centroid, which is the point of intersection of the three medians, each of which is divided in a 2:1 ratio by the centroid. please visit also BenefitsCal App
+11 To solve this problem, we can use the concept of permutations with repetition. We have two groups of beads: one group with two identical beads and another group with four identical beads. Let's represent the two groups as "A" and "B", respectively. Then, we can use the following formula:
n! / (n1! x n2! x ... nk!) www.c4yourself.com
where n is the total number of elements (beads in this case), and n1, n2, ..., nk represent the number of elements in each group. In our case, we have:
n = 6 n1 = 2 (two identical beads) n2 = 4 (four identical beads)
Using the formula, we get:
6! / (2! x 4!) = 15
Therefore, Joanna can assemble her bracelet in 15 different ways. Note that we divide by 2! and 4! to account for the fact that the beads within each group are identical, and we divide by the product of these factorials to account for the fact that the groups themselves are indistinguishable. Finally, we divide by the total number of permutations (6!) to account for the fact that two identical bracelets are considered identical.
+11 hello
To find two numbers that add to give -9 and multiply to give 18, we can use algebraic equations.
Let's call the two numbers we're looking for "x" and "y". Then we can write:
x + y = -9 (equation 1)
xy = 18 (equation 2)
We can solve equation 1 for one of the variables. For example, we can solve for "x" by subtracting "y" from both sides:
x = -9 - y
We can then substitute this expression for "x" into equation 2, giving:
(-9 - y)y = 18
Expanding the left-hand side and rearranging, we get: Nexus Iceland App
y^2 + 9y - 18 = 0
This is a quadratic equation that we can solve using the quadratic formula:
y = (-9 ± sqrt(9^2 - 4(1)(-18))) / (2(1))
Simplifying:
y = (-9 ± 15) / 2
So we have two possible values for "y":
y = -12 or y = 3/2
If y = -12, then we can use equation 1 to find x:
x + (-12) = -9
x = 3
So one possible pair of numbers that add to give -9 and multiply to give 18 is 3 and -12.
If y = 3/2, then we can use equation 1 to find x:
x + (3/2) = -9
x = -21/2
So another possible pair of numbers that add to give -9 and multiply to give 18 is -21/2 and 3/2.
Therefore, there are two possible pairs of numbers that add to give -9 and multiply to give 18: {3, -12} and {-21/2, 3/2}.