n+3P6 : n+2P4 = 14 : 1. Find 'n' .
n+3P6 = (n+3)(n+2)(n+1)(n)(n-1)(n-2)
n+2P4 = (n+2)(n+1)(n)(n-1)
(n+3)(n-2) = 14
n=4
To solve for 'n' in the equation n + 3P6 : n + 2P4 = 14 : 1, where P represents permutation, we need to use the concept of permutations and apply it step by step.
Step 1: Determine the number of permutations
In the equation, the number after the 'P' represents the number of objects being permuted. In this case, we have 6 objects being permuted in the numerator (3P6) and 4 objects being permuted in the denominator (2P4).
The formula for permutations is given by nPr = n! / (n - r)!, where n is the total number of objects, and r is the number of objects being permuted.
For 3P6, we have n = 6 and r = 3. Plugging these values into the formula, we get:
3P6 = 6! / (6 - 3)! = 720 / 6 = 120
For 2P4, we have n = 4 and r = 2. Plugging these values into the formula, we get:
2P4 = 4! / (4 - 2)! = 24 / 2 = 12
Step 2: Rewrite the equation using the number of permutations
Now, replace 3P6 with 120 and 2P4 with 12 in the equation:
n + 120 : n + 12 = 14 : 1
Step 3: Solve for 'n'
To solve for 'n', we need to find the value that satisfies the equation. We can do this by cross-multiplying.
Cross-multiplying gives:
( n + 120 ) * 1 = ( n + 12 ) * 14
Simplifying the equation:
n + 120 = 14 * ( n + 12 )
n + 120 = 14n + 168
Subtracting 'n' from both sides:
120 = 13n + 168
Subtracting 168 from both sides:
120 - 168 = 13n
-48 = 13n
Dividing both sides by 13:
-48 / 13 = n
Therefore, the value of 'n' that satisfies the equation is approximately -3.6923.