Complex Numbers
Complex Numbers
star_batch_jee_advanced_2025
Grade 11
Question:
Complex numbers $z_1, z_2, z_3$ and $z_4$ correspond to the points $A, B, C$ and $D$, respectively, on a circle $|z| = 1$. If $z_1 + z_2 + z_3 + z_4 = 0$, then $ABCD$ is necessarily
a triangle
a square
a rhombus
a parallelogram
Step-by-Step Solution
Key Concept: The difference between total selections and selections excluding one element equals selections of one fewer element.
Step 1: Calculate the total number of ways to select 5 students from 25.
The total number of ways to select 5 students from 25 available students is given by the combination formula $\binom{n}{k} = \frac{n!}{k!(n-k)!}$.
$$ \text{Total selections} = \binom{25}{5} = ^{25}C_5 $$
Step 2: Calculate the number of ways to select 5 students such that a particular student is included.
If a particular student is already selected, we need to choose the remaining 4 students from the remaining 24 students.
$$ \text{Selections including a particular student} = \binom{24}{4} = ^{24}C_4 $$
Step 3: Calculate the difference between the total selections and selections including a particular student.
The difference between the total number of selections and the number of selections including a particular student is calculated. This difference corresponds to the number of ways to select 5 students where the particular student is *not* included.
Using the identity $\binom{n}{k} - \binom{n-1}{k-1} = \binom{n-1}{k}$, we have:
$$ ^{25}C_5 - ^{24}C_4 = ^{24}C_5 $$
This represents the number of ways to select 5 students from 25 such that a specific student is *not* chosen.
The final answer is $\boxed{^{24}C_5}$.
Correct Answer: 1,4