Practice Questions
How many ways can the letters of the word 'BANANA' be arranged?
Total letters = 6. Repeated: A appears 3 times, N appears 2 times. Ways = $\frac{6!}{3! \times 2!} = \frac{720}{12} = 60$.
In how many ways can 5 people be seated in a row?
Arranging $n$ distinct objects in a row = $n! = 5! = 120$.
How many ways can a committee of 3 be formed from 8 people?
$8C3 = \frac{8!}{3! \times 5!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56$.
How many handshakes occur if 6 people each shake hands with every other person exactly once?
Handshakes = $nC2 = \frac{6 \times 5}{2} = 15$.
In how many different ways can the letters of the word MANAGER be arranged?
MANAGER: M=1, A=2, N=1, G=1, E=1, R=1. Total = 7, A repeated twice. Arrangements = $7! / 2! = 5040 / 2 = 2520$.
In how many ways can 4 boys and 3 girls be seated in a row such that no two girls sit together?
Arrange 4 boys in $4! = 24$ ways. There are 5 gaps. Choose 3 gaps and arrange girls: $5P_3 = 60$. Total = $24 \times 60 = 1440$.
How many 5-digit numbers can be formed from digits 1–7 without repetition and divisible by 5?
Divisible by 5 means last digit must be 5. Remaining 4 digits chosen from 1,2,3,4,6,7: $6P_4 = 6 \times 5 \times 4 \times 3 = 360$.
In how many ways can a President, Vice-President and Secretary be selected from 8 people?
Order matters. $8P_3 = 8 \times 7 \times 6 = 336$.
In how many ways can letters of SUCCESS be arranged such that vowels come together?
Vowels: U, E (2 distinct). Treat as unit: (UE), S, C, C, S, S = 6 units with S=3, C=2. Arrangements = $6!/(3!2!) \times 2! = 60 \times 2 = 120$.
From 10 students, how many ways can a team of 5 be formed if 2 particular students refuse to work together?
Total teams = $10C5 = 252$. Teams with both = $8C3 = 56$. Teams without both = $252 - 56 = 196$.
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