Sequence and SeriesMCQMTP Apr 21Question 1845 of 212
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Given: P(7,k)=60P(7,k−3)\displaystyle P(7, k) = 60 P(7, k-3). Then: k=\displaystyle k =

Options

Ak=9\displaystyle k=9
Bk=8\displaystyle k=8
Ck=5\displaystyle k=5
Dk=0\displaystyle k=0
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Correct Answer

✅ Option c — k=5\displaystyle k=5

All Options:

  • Ak=9\displaystyle k=9
  • Bk=8\displaystyle k=8
  • Ck=5\displaystyle k=5
  • Dk=0\displaystyle k=0

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Detailed Solution & Explanation

Using the permutation formula P(n,r)=n!(n−r)!\displaystyle P(n, r) = \frac{n!}{(n-r)!}:
P(7,k)=60⋅P(7,k−3)P(7, k) = 60 \cdot P(7, k-3)
7!(7−k)!=60⋅7!(10−k)!\frac{7!}{(7-k)!} = 60 \cdot \frac{7!}{(10-k)!}
(10−k)!=60(7−k)!(10-k)! = 60 (7-k)!
(10−k)(9−k)(8−k)(7−k)!=60(7−k)!(10-k)(9-k)(8-k)(7-k)! = 60(7-k)!
(10−k)(9−k)(8−k)=60(10-k)(9-k)(8-k) = 60

Since 60=5⋅4⋅3\displaystyle 60 = 5 \cdot 4 \cdot 3 (product of three consecutive integers):
10−k=5  ⟹  k=510-k = 5 \implies k = 5
Hence, **Option C** is the correct answer.

About This Chapter: Sequence and Series

Paper

Paper 3: Quantitative Aptitude

Weightage

4-6 Marks

Key Topics

Arithmetic & Geometric Progressions

This chapter covers Arithmetic Progressions (AP) and Geometric Progressions (GP). Students learn how to find the nth term, sum of n terms, arithmetic/geometric means, and sum to infinity of a GP.

View Official ICAI Syllabus

Exam Strategy Tip

For complex 'sum of series' questions, a great hack is to substitute n = 1 and n = 2 into the question and the options to see which one matches, completely bypassing the formula.

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