1. You are setting up a system of programming codes for a universal TV remote. Each code will have five digits and cannot include a or repeat any digits. How many codes a) Do not contain a 7 ? b) Have 3 as the third digit and 5 as the fifth digit? c) Begin with an even number? d) Begin and end with an even number?

To solve the problem, we will analyze each part step by step, considering the constraints provided. ### Known Conditions: - Each code consists of 5 digits. - Digits cannot include 'a' (which I assume means the digit '0' since 'a' is not a digit). - Digits cannot repeat. ### a) Codes that do not contain a 7 The available digits are 0, 1, 2, 3, 4, 5, 6, 8, 9 (9 digits total, excluding 7). 1. **First digit**: Can be any of the 9 digits (0-6, 8, 9). 2. **Second digit**: Can be any of the remaining 8 digits. 3. **Third digit**: Can be any of the remaining 7 digits. 4. **Fourth digit**: Can be any of the remaining 6 digits. 5. **Fifth digit**: Can be any of the remaining 5 digits. The total number of codes is calculated as: \[ 9 \times 8 \times 7 \times 6 \times 5 \] ### b) Codes with 3 as the third digit and 5 as the fifth digit 1. **Third digit**: Fixed as 3. 2. **Fifth digit**: Fixed as 5. 3. **First digit**: Can be any of the remaining 8 digits (0, 1, 2, 4, 6, 7, 8, 9). 4. **Second digit**: Can be any of the remaining 7 digits. 5. **Fourth digit**: Can be any of the remaining 6 digits. The total number of codes is calculated as: \[ 8 \times 7 \times 6 \] ### c) Codes that begin with an even number The available even digits are 0, 2, 4, 6, 8 (5 options). 1. **First digit**: Can be any of the 5 even digits. 2. **Second digit**: Can be any of the remaining 9 digits. 3. **Third digit**: Can be any of the remaining 8 digits. 4. **Fourth digit**: Can be any of the remaining 7 digits. 5. **Fifth digit**: Can be any of the remaining 6 digits. The total number of codes is calculated as: \[ 5 \times 9 \times 8 \times 7 \times 6 \] ### d) Codes that begin and end with an even number 1. **First digit**: Can be any of the 5 even digits. 2. **Fifth digit**: Can be any of the remaining 4 even digits (since it cannot repeat the first digit). 3. **Second digit**: Can be any of the remaining 8 digits. 4. **Third digit**: Can be any of the remaining 7 digits. 5. **Fourth digit**: Can be any of the remaining 6 digits. The total number of codes is calculated as: \[ 5 \times 4 \times 8 \times 7 \times 6 \] Now, let's calculate the total number of codes for each part. ### Calculations I will perform the calculations for each part now. Calculate the value by following steps: - step0: Calculate: \(8\times 7\times 6\) - step1: Multiply the terms: \(56\times 6\) - step2: Multiply the numbers: \(336\) Calculate or simplify the expression \( 5*9*8*7*6 \). Calculate the value by following steps: - step0: Calculate: \(5\times 9\times 8\times 7\times 6\) - step1: Multiply the terms: \(45\times 8\times 7\times 6\) - step2: Multiply the terms: \(360\times 7\times 6\) - step3: Multiply the terms: \(2520\times 6\) - step4: Multiply the numbers: \(15120\) Calculate or simplify the expression \( 5*4*8*7*6 \). Calculate the value by following steps: - step0: Calculate: \(5\times 4\times 8\times 7\times 6\) - step1: Multiply the terms: \(20\times 8\times 7\times 6\) - step2: Multiply the terms: \(160\times 7\times 6\) - step3: Multiply the terms: \(1120\times 6\) - step4: Multiply the numbers: \(6720\) Calculate or simplify the expression \( 9*8*7*6*5 \). Calculate the value by following steps: - step0: Calculate: \(9\times 8\times 7\times 6\times 5\) - step1: Multiply the terms: \(72\times 7\times 6\times 5\) - step2: Multiply the terms: \(504\times 6\times 5\) - step3: Multiply the terms: \(3024\times 5\) - step4: Multiply the numbers: \(15120\) Here are the results for each part of the problem: ### a) Codes that do not contain a 7 The total number of codes is \( 15120 \). ### b) Codes with 3 as the third digit and 5 as the fifth digit The total number of codes is \( 336 \). ### c) Codes that begin with an even number The total number of codes is \( 15120 \). ### d) Codes that begin and end with an even number The total number of codes is \( 6720 \). These calculations provide the number of valid codes for each specified condition.