How many 4 digit numbers can be formed from the digits 1, 1, 2, 2, 3, 3, 4 and 5?

Miscellaneous Exercise · Q4

Q. How many 4 digit numbers can be formed from the digits 1, 1, 2, 2, 3, 3, 4 and 5?

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Concept understanding — Miscellaneous

The Concept of "Miscellaneous" – First Principles

You've seen the word before, probably in a file folder or a drawer: "Miscellaneous." It's the catch-all, the "everything else" bin. In mathematics and exam contexts, it means exactly that — but with a precise, deliberate structure.

Intuition First

Imagine you're sorting a box of objects. You have a pile for "red balls," a pile for "blue cubes," a pile for "green triangles." But there are a few objects that don't fit neatly: a yellow sphere, a striped cylinder, a tiny purple star. Where do they go? You create a new pile labelled "Miscellaneous." This pile isn't random — it's the set of everything that didn't belong to any of the specific categories you already defined.

In mathematics, "Miscellaneous" works the same way. It's a collection of items that don't fall under the main headings you've already listed, but are still part of the overall set you're considering.

The Precise Statement

Formally, when a problem or a chapter lists a section called "Miscellaneous," it means:

Miscellaneous = The set of all elements (or problems, or cases) that belong to the domain under discussion but are not covered by any of the explicitly named categories or sub-sections.

In exam contexts — especially in Indian board exams like CBSE — "Miscellaneous" appears in two main ways:

Miscellaneous Exercises at the end of a chapter. These problems test the entire chapter's concepts, often mixing ideas from different sections. They are not "extra" or "optional" — they are comprehensive.
Miscellaneous Examples within a chapter. These illustrate applications that don't fit neatly into a single sub-topic, or that combine multiple concepts.

Important

In exam preparation, "Miscellaneous" problems are not less important. They often appear in board exams because they test integration of concepts — exactly what examiners want to see.

Why It Matters

The power of "Miscellaneous" is that it forces you to think beyond rigid categories. A problem that asks you to find the area under a curve using integration, but also involves a trigonometric substitution and a limit — that's a miscellaneous problem. It doesn't belong to "Integration by Substitution" alone, nor to "Trigonometric Integrals" alone. It belongs to the intersection of several techniques.

When you encounter a "Miscellaneous" section, treat it as a synthesis challenge. It's the exam's way of saying: "Now that you've learned the pieces, show me you can put them together." …

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The 8 available digits are $1,1,2,2,3,3,4,5$ — digits $1,2,3$ can each appear at most twice, $4$ and $5$ at most once — so split the count into cases by how many digits repeat in the 4-digit number.

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Case A — all four digits distinct. Choose any 4 distinct digit-values out of the 5 available values $\{1,2,3,4,5\}$ : $^5C_4=5$ ways. Arrange the 4 distinct digits: $4!=24$ ways. Sub-total $=5\times24=120$ .
Case B — exactly one digit repeated twice, other two digits distinct. The repeated digit must be one that has 2 copies available, i.e. from $\{1,2,3\}$ : $3$ choices. Choose the other 2 (distinct, non-repeated) digit-values from the remaining 4 values: $^4C_2=6$ ways. Arrange the multiset $\{x,x,y,z\}$ (one pair + two singles) in $\dfrac{4!}{2!}=12$ ways. Sub-total $=3\times6\times12=216$ . …

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Q.How many 4 digit numbers can be formed from the digits 1, 1, 2, 2, 3, 3, 4 and 5?

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