MAH CET MCA Permutations And Combinations Previous Year Questions (PYQs)
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Let $K$ be the set of all odd numbers less than $200$ and $M$ be the set of products of two distinct numbers taken from $K$, then find the total number which is divisible by $5$ in $M$.
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Solution
Among $100$ odd numbers, $20$ are multiples of $5$.
Required count $= 100 \times 20 - C(20,2) = 392$.
The number of diagonals in a polygon of $m$ sides is
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Solution
Formula for number of diagonals $= \dfrac{m(m-3)}{2}$.
A five digit number divisible by 3 using digits 0, 1, 2, 3, 4, 5 is to be made
without repetition. Find of such word.
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Solution
Digits available: {0,1,2,3,4,5}.
Sum of all six = 15 divisible by 3.
To make a 5-digit number divisible by 3, exclude one digit whose value 0, i.e., exclude 0 or 3.
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Exclude 0 → use {1,2,3,4,5}: all 5! = 120 numbers (no leading-zero issue).
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Exclude 3 → use {0,1,2,4,5}: total 5! − 4! (leading 0) = 120 − 24 = 96.
Total = 120 + 96 = 216
Find the total number of ways to put $20$ balls into $5$ boxes so that the first box contains only one ball.
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Solution
Choose which ball goes to Box 1: $20$ ways.
Each of the remaining $19$ balls can go to any of Boxes $2$–$5$: $4^{19}$ ways.
Total $=20\cdot4^{19}$.
Find the number of arrangements of the letters of the word $,\text{INDEPENDENCE},$ such that
(i) the words start with $I$ and end with $P$.
(ii) all the vowels never occur together.
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Solution
Letters in INDEPENDENCE: $E^4,\ N^3,\ D^2,\ I,\ P,\ C$ (total $12$).
(i) Fix $I$ at start and $P$ at end. Arrange the remaining $10$ symbols: $E^4,\ N^3,\ D^2,\ C$.
Number $= \dfrac{10!}{4!,3!,2!}=12600$.
(ii) Total arrangements $= \dfrac{12!}{4!,3!,2!}=1663200$.
“All vowels together”: treat $(\text{vowels}=E^4I)$ as one block. Then items are
${\text{block}, N^3, D^2, C, P}$ → $8$ items with repeats $N^3, D^2$.
Arrangements of items $= \dfrac{8!}{3!,2!}=3360$.
Internal vowel-block arrangements $= \dfrac{5!}{4!}=5$.
Together $= 3360\times5=16800$.
Hence “never together” $=1663200-16800=1646400$.
Straight lines are drawn by joining m points on a straight line to n points on another line. Then excluding the given points, the number of point of intersection of the lines drawn is (no two lines drawn are parallel and no three lines are concurrent).
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Solution
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