Permutations and CombinationsPYQ Jan 26Question 4561 of 183
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If nCr−1=28,nCr=56,nCr+1=70\displaystyle ^nC_{r-1} = 28, ^nC_r = 56, ^nC_{r+1} = 70, then the value of n and r are

Options

An = 8, r = 3
Bn = 8, r = 4
Cn = 9, r = 4
Dn = 9, r = 3
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Correct Answer

✅ Option a — n = 8, r = 3

All Options:

  • An = 8, r = 3
  • Bn = 8, r = 4
  • Cn = 9, r = 4
  • Dn = 9, r = 3

Detailed Solution & Explanation

We are given the following combinations: nCr−1=28— (1)^nC_{r-1} = 28 \quad \text{--- (1)} nCr=56— (2)^nC_r = 56 \quad \text{--- (2)} nCr+1=70— (3)^nC_{r+1} = 70 \quad \text{--- (3)}
We use the division formula for combinations: nCknCk−1=n−k+1k\frac{^nC_k}{^nC_{k-1}} = \frac{n-k+1}{k}
Dividing equation (2) by equation (1): nCrnCr−1=5628=2\frac{^nC_r}{^nC_{r-1}} = \frac{56}{28} = 2 n−r+1r=2  ⟹  n−r+1=2r  ⟹  n−3r+1=0— (Eq I)\frac{n - r + 1}{r} = 2 \implies n - r + 1 = 2r \implies n - 3r + 1 = 0 \quad \text{--- (Eq I)}
Dividing equation (3) by equation (2): nCr+1nCr=7056=54\frac{^nC_{r+1}}{^nC_r} = \frac{70}{56} = \frac{5}{4} n−rr+1=54  ⟹  4(n−r)=5(r+1)  ⟹  4n−4r=5r+5  ⟹  4n−9r−5=0— (Eq II)\frac{n - r}{r + 1} = \frac{5}{4} \implies 4(n - r) = 5(r + 1) \implies 4n - 4r = 5r + 5 \implies 4n - 9r - 5 = 0 \quad \text{--- (Eq II)}
From (Eq I), we express n\displaystyle n in terms of r\displaystyle r: n=3r−1n = 3r - 1 Substitute n=3r−1\displaystyle n = 3r - 1 into (Eq II): 4(3r−1)−9r−5=04(3r - 1) - 9r - 5 = 0 12r−4−9r−5=012r - 4 - 9r - 5 = 0 3r−9=0  ⟹  r=33r - 9 = 0 \implies r = 3
Substituting r=3\displaystyle r = 3 back into the expression for n\displaystyle n: n=3(3)−1=8n = 3(3) - 1 = 8 Thus, n=8\displaystyle n = 8 and r=3\displaystyle r = 3. Hence, **Option A** is the correct answer.

About This Chapter: Permutations and Combinations

Paper

Paper 3: Quantitative Aptitude

Weightage

4-6 Marks

Key Topics

Factorials, Permutations, Combinations

This chapter deals with the fundamental principles of counting. It covers factorials, circular permutations, restricted permutations, combinations, and the differences between selecting items versus arranging them.

View Official ICAI Syllabus

Exam Strategy Tip

The most common mistake is confusing 'P' (Arrangement) with 'C' (Selection). If order matters (like opening a lock), use P. If order doesn't matter (like choosing a team), use C.

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