Phrases Previous Year Questions (PYQs) – Permutations And Combinations – Page 1 of 4

Total bottles and boxes: 9 each, labeled 1 to 9.

We are asked to count permutations of bottles such that exactly 5 bottles go into their own numbered boxes.

Step 1: Choose 5 positions to be fixed points (i.e., bottle number matches box number).

Number of ways = $(\genfrac{}{}{0ex}{}{9}{5})$

Step 2: Remaining 4 positions must be a derangement (no bottle goes into its matching box).

Let $D_4$ be the number of derangements of 4 items.

$D_4=9$

Step 3: Total ways = $(\genfrac{}{}{0ex}{}{9}{5})\times D_4=126\times 9=1134$

✅ Final Answer: $\boxed{{\displaystyle 1134}}$

Question: Find the value of:

$$\sum _{r=1}^n\frac{n{P}_r}{r!}$$

Solution:

We know: $n{P}_r=\frac{n!}{(n-r)!}\Rightarrow \frac{n{P}_r}{r!}=\frac{n!}{(n-r)!\cdot r!}=(\genfrac{}{}{0ex}{}{n}{r})$

Therefore,

$$\sum _{r=1}^n\frac{n{P}_r}{r!}=\sum _{r=1}^n(\genfrac{}{}{0ex}{}{n}{r})=2^n-1$$

Final Answer: $$\boxed{{\displaystyle 2^n-1}}$$

Total Number of Intersection Points

Given:

• 10 distinct lines: $L_1,L_2,\dots ,L_{10}$
• $L_2,L_4,L_6,L_8,L_{10}$: parallel (no intersections among them)
• $L_1,L_3,L_5,L_7,L_9$: concurrent at point $C$ (intersect at one point)

? Calculation:

$$\text{Total line pairs: }(\genfrac{}{}{0ex}{}{10}{2})=45$$

$$\text{Subtract parallel pairs: }(\genfrac{}{}{0ex}{}{5}{2})=10\Rightarrow 45-10=35$$

$$\text{Concurrent at one point: reduce }10\text{ pairs to 1 point}\Rightarrow 35-9=\boxed{{\displaystyle 26}}$$

✅ Final Answer: $\boxed{{\displaystyle 26}}$ unique points of intersection