**1 mark** for listing at least 4 correct numbers. M1\n* **2 marks** for listing at least 8 correct numbers. M1\n* **3 marks** for finding all 12 correct numbers. A1"}]},{"id":"cf757509-ba50-4e28-beea-7413d08edc00:tree-diagram","label":"Tree Diagram","steps":[{"type":"text","order":0,"title":"Understand the cards","content":"We are given four number cards:\n$$\\text{Cards: } 5, \\, 2, \\, 5, \\, 9$$\n\nTo find all the different possible four-digit numbers, we can draw a **tree diagram** branch by branch. Let's list our available cards: we have one **$2$**, two **$5$**s, and one **$9$**.","highlights":[{"text":"5, 2, 5, 9","location":"specification"}]},{"type":"text","order":1,"title":"Branch 1: Thousands digit is 2","content":"Let's start our tree diagram with the **Thousands** place. \n\nIf we choose **$2$** as the thousands digit, our remaining cards for the hundreds, tens, and units are $5, 5, 9$:\n\n* **Thousands:** $2$\n * **Hundreds:** can be $5$ or $9$.\n * If **Hundreds** is $5$, the remaining cards are $5$ and $9$:\n * **Tens:** $5$ $\\rightarrow$ **Units:** $9$ (Number: **$2559$**)\n * **Tens:** $9$ $\\rightarrow$ **Units:** $5$ (Number: **$2595$**)\n * If **Hundreds** is $9$, the remaining cards are $5$ and $5$:\n * **Tens:** $5$ $\\rightarrow$ **Units:** $5$ (Number: **$2955$**)\n\nThis first branch gives us **$3$** unique numbers starting with $2$:\n$$\\{2559, \\, 2595, \\, 2955\\}$$"},{"type":"text","order":2,"title":"Branch 2: Thousands digit is 5","content":"Next, let's look at the tree branch if the **Thousands** digit is **$5$**.\n\nThe remaining cards for the hundreds, tens, and units are $2, 5, 9$:\n\n* **Thousands:** $5$\n * **Hundreds:** can be $2$, $5$, or $9$.\n * If **Hundreds** is $2$, the remaining cards are $5$ and $9$:\n * **Tens:** $5$ $\\rightarrow$ **Units:** $9$ (Number: **$5259$**)\n * **Tens:** $9$ $\\rightarrow$ **Units:** $5$ (Number: **$5295$**)\n * If **Hundreds** is $5$, the remaining cards are $2$ and $9$:\n * **Tens:** $2$ $\\rightarrow$ **Units:** $9$ (Number: **$5529$**)\n * **Tens:** $9$ $\\rightarrow$ **Units:** $2$ (Number: **$5592$**)\n * If **Hundreds** is $9$, the remaining cards are $2$ and $5$:\n * **Tens:** $2$ $\\rightarrow$ **Units:** $5$ (Number: **$5925$**)\n * **Tens:** $5$ $\\rightarrow$ **Units:** $2$ (Number: **$5952$**)\n\nThis second branch gives us **$6$** unique numbers starting with $5$:\n$$\\{5259, \\, 5295, \\, 5529, \\, 5592, \\, 5925, \\, 5952\\}$$"},{"type":"text","order":3,"title":"Branch 3: Thousands digit is 9","content":"Finally, let's look at the tree branch if the **Thousands** digit is **$9$**.\n\nThe remaining cards are $2, 5, 5$:\n\n* **Thousands:** $9$\n * **Hundreds:** can be $2$ or $5$.\n * If **Hundreds** is $2$, the remaining cards are $5$ and $5$:\n * **Tens:** $5$ $\\rightarrow$ **Units:** $5$ (Number: **$9255$**)\n * If **Hundreds** is $5$, the remaining cards are $2$ and $5$:\n * **Tens:** $2$ $\\rightarrow$ **Units:** $5$ (Number: **$9525$**)\n * **Tens:** $5$ $\\rightarrow$ **Units:** $2$ (Number: **$9552$**)\n\nThis third branch gives us **$3$** unique numbers starting with $9$:\n$$\\{9255, \\, 9525, \\, 9552\\}$$"},{"type":"text","order":4,"title":"Compile the complete list","content":"By tracing all the branches of our tree diagram, we find a total of $3 + 6 + 3 = 12$ unique four-digit numbers:\n\n* **Starting with 2:** $2559, \\, 2595, \\, 2955$\n* **Starting with 5:** $5259, \\, 5295, \\, 5529, \\, 5592, \\, 5925, \\, 5952$\n* **Starting with 9:** $9255, \\, 9525, \\, 9552$\n\n### Markscheme Info\n* Award **1 mark** for writing down at least 4 correct numbers. M1\n* Award **2 marks** for writing down at least 8 correct numbers. M1\n* Award **3 marks** for listing all 12 correct numbers. A1"}]}],"generatedAt":"2026-06-12T17:11:18.698Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null}],"subjectCode":"gcse_math

Systematic Listing

Question 26

Systematic Listing Questions

Systematic Listing

Question 26

Systematic Listing Questions