Given, that \(4P4_5 = 119_{10}\), find the value of P

To solve this problem, we need to first understand what \(4P4_5\) means. Here, 4P4 is a permutation, which means we are choosing 4 items out of 4, and arranging them in a particular order. The subscript 5 means we are working in base 5. We can calculate the value of \(4P4_5\) as follows: \(4P4_5 = 4! = 4\times3\times2\times1 = 24\) Now, we are given that \(4P4_5 = 119_{10}\), which means we need to convert 119 to base 5 to find the value of P. 119 in base 5 is equal to 1444. So, we can write the equation: \(4P4_5 = 119_{10} = 1444_5\) To find the value of P, we can write out the permutation using P instead of 4P4: \(P\times(P-1)\times(P-2)\times(P-3)_5 = 1444_5\) We can start by trying to divide 1444 by 3, since 3 is a factor of 24 (the value of 4P4). \(1444_5 \div 3 = 434_5\) So, we can write: \(P\times(P-1)\times(P-2)\times(P-3)_5 = 434_5\times3\) Now, we can see that 434 is divisible by 4, since the last two digits (34) are divisible by 4. \(434_5 \div 4 = 104_5\) So, we can write: \(P\times(P-1)\times(P-2)\times(P-3)_5 = 104_5\times3\times4\) Simplifying, we get: \(P\times(P-1)\times(P-2)\times(P-3)_5 = 4992_5\) We can see that the last digit is 2, which means P-3 must be equal to 2. So, P = 5 - 2 = 3. Therefore, the value of P is 3.