https://file.aiquickdraw.com/mathbot/file/17297527875151wzd9ii4.jpg

Let's go through the questions in the image step by step:

### Q1: \( nC_4 = 5 \times nC_3 \), find \( n \)
- Use the combination formula \( nC_r = \frac{n!}{r!(n-r)!} \).
- We have \( nC_4 = \frac{n!}{4!(n-4)!} \) and \( nC_3 = \frac{n!}{3!(n-3)!} \).
- Setting \( nC_4 = 5 \times nC_3 \), solve for \( n \).

### Q2: \( nC_r = 6720 \), \( nC_{r+1} = 56 \); find \( n \) and \( r \)
- Use the relation \( nC_{r+1} = \frac{n-r}{r+1} nC_r \) to find \( n \) and \( r \).

### Q3: \( 18C_r = 18C_{r+2} \); find \( C_6 \)
- Use the identity \( nC_r = nC_{n-r} \) to simplify and find \( r \).
- Then compute \( C_6 \).

### Q4: \( 12C_r = 12C_{r+2} \); find \( C_3 \)
- Again, use symmetry in combinations to find \( r \), then compute \( C_3 \).

### Q5: \( 12C_8 = 12C_{r-2} \); find \( C_3 \)
- Use the combination formula and symmetry to find \( r \) and then \( C_3 \).

Would you like detailed solutions for these?

Various combinatorics problems involving nCr (binomial coefficient). Problems include finding values of n, r, and specific combinations.

Learn how to solve a variety of combinatorics problems involving binomial coefficients (nCr). These problems focus on using combination formulas, the binomial theorem, and symmetry properties to find unknown values of n and r. Suitable for advanced high school or early college students.

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