Question

Five candidates are contesting an election, and three members are to be elected. A voter can vote for any number of candidates, but not more than the number of members to be elected. The number of ways a person can cast their vote is :

• A. \( 5 \)
• B. \( 15 \)
• C. \( 20 \)
• D. \( 25 \) Correct Answer: (D) \( 25 \) Solution:

Hint:

Always read carefully if the question allows for a "zero" vote (abstaining). If it did, the answer would be \( \sum_{r=0}^{3} ^5C_r = 1 + 25 = 26 \).
Also, for small values like \( n=5 \), sketching the Pascal's triangle row \( (1, 5, 10, 10, 5, 1) \) is a very fast way to get the values for \( ^5C_r \).

Answer 1

The Correct Option is D

Step 1: Decide the choices a voter has.
There are $5$ candidates and at most $3$ can be picked. A voter may vote for $1$, $2$, or $3$ of them. Order does not matter, so we use combinations.

Step 2: Count voting for one or two.
\[ \,^5C_1 = 5, \qquad \,^5C_2 = \frac{5\cdot 4}{2} = 10 \]

Step 3: Count voting for three.
\[ \,^5C_3 = \,^5C_2 = 10 \]

Step 4: Add them up.
\[ 5 + 10 + 10 = 25 \]
So there are $25$ ways, which is option 4.
\[ \boxed{25} \]