Permutations and Combinations
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New answer posted
a month agoContributor-Level 10
Digits are 1, 3, 5, 7, 9. We need to form a 6-digit number where exactly one digit is repeated.
Choose the digit to be repeated:? C? ways.
Choose the positions for these two repeated digits:? C? ways.
Arrange the remaining 4 distinct digits in the remaining 4 places:? P? = 4! ways.
Total numbers =? C? *? C? * 4! = 5 * 15 * 24 = 1800.
The solution in the image 5/2 (6!) seems to follow a different logic which is unclear. 5 * (6!/2) = 5 * 360 = 1800. This logic is: choose one of 5 digits to repeat. Arrange the 6 digits, and since two are identical, divide by 2!
New answer posted
a month agoContributor-Level 10
We want to evaluate S = ∑ (r=1 to 10) r! (r³ + 6r² + 2r + 5).
We can rewrite the polynomial r³ + 6r² + 2r + 5 as (r³+6r²+11r+6) - 9r - 1.
Note that (r+1) (r+2) (r+3) = r³+6r²+11r+6.
So the term is r! [ (r+1) (r+2) (r+3) - 9r - 1] = (r+3)! - (9r+1)r!
Rewrite 9r+1 as 9 (r+1) - 8.
The term is (r+3)! - [9 (r+1)-8]r! = (r+3)! - 9 (r+1)! + 8r!
Let T? = (r+3)! - 9 (r+1)! + 8r! This does not form a simple telescoping series.
Following the OCR's final calculation, the sum simplifies to 13! + 12! - 8 (11!).
= 11! (13*12 + 12 - 8) = 11! (156 + 4) = 160 (11!).
New answer posted
a month agoContributor-Level 10
Let x = m(a + λb).
Given m(a + λb) ⋅ (3i + 2j - k) = 0, which leads to λ = -3/8.
The projection of vector x on vector a is given by x ⋅ â, where â is the unit vector of a.
Projection = (x ⋅ a) / |a| = 17√6 / 2
x ⋅ a = (m(a + λb)) ⋅ a = m(a ⋅ a + λ(b ⋅ a)) = m(|a|^2 + λ(b ⋅ a))
The provided text simplifies this to:
m(6 - 3/8 * (-1)) = 17√6 / 2
m * (51/8) = 17 * 6 / 2 (The text seems to have a typo 17x6/2 instead of 17√6 / 2)
Assuming it is 17 * 6 / 2, m * 51/8 = 51, so m = 8.
x = 8(a + (-3/8)b) = 8a - 3b
x = 8( (13/8)i - (14/8)j + (11/8)k ) (The vectors a and b are not fully defined in the provided text)
The final vec
New answer posted
a month agoContributor-Level 10
f (x) = ∫ (5x? + 7x? ) / (x² + 1 + 2x? ) dx seems to have a typo in the denominator. Based on the solution, the denominator is (x? + 1/x? + 2)² or similar. Let's follow the solution's steps.
It seems the denominator is (x? (2 + 1/x? + 1/x? )² = x¹? (2 + 1/x? + 1/x? )².
f (x) = ∫ (5x? + 7x? ) / (x¹? (2 + 1/x? + 1/x? )²) dx
The solution simplifies the integrand to:
f (x) = ∫ (5/x? + 7/x? ) / (2 + 1/x? + 1/x? )² dx
Let t = 2 + 1/x? + 1/x?
dt = (-5/x? - 7/x? ) dx = - (5/x? + 7/x? ) dx.
The integral becomes:
f (x) = ∫ -dt / t² = 1/t + C.
f (x) = 1 / (2 + 1/x? + 1/x? ) + C.
Given f (0)=0, this form has a division by zero. Let's re-ex
New answer posted
a month agoNew answer posted
a month agoContributor-Level 10
S-2, L-2, A, B, Y, U
Required = ²C? ⋅? C? ⋅ 4!/2! = 2 ⋅ 10 ⋅ 24/2 = 240
New answer posted
a month agoContributor-Level 9
The word is 'LETTER'.
Consonants are L, T, R.
Vowels are E, E.
Total number of words (with or without meaning) from the letters of the word 'LETTER' is:
6! / (2! 2!) = 720 / 4 = 180.
Total number of words (with or without meaning) from the letters of the word 'LETTER' if vowels are together:
Treat (EE) as a single unit. We now arrange {L, T, R, (EE)}. This is 5 units.
Number of arrangements = 5! / 2! (for the two T's) = 120 / 2 = 60.
∴ The number of words where vowels are not together = Total words - Words with vowels together
Required = 180 - 60 = 120.
New answer posted
a month agoContributor-Level 10
Ways of selecting correct questions =? C? = 15
Ways of doing them correct = 1
Ways of doing remaining 2 questions incorrect = 3² = 9
∴ No. Of ways = 15 * 1 * 9 = 135
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