Counting Questions with Solutions — Set 2 (Q11-Q20)

This set contains counting questions with solutions — Set 2 (Q11-Q20) covering a mix of question types and difficulty levels — from basic to advanced — exactly as asked in real competitive exams.

Solutions are written in a simple, step-by-step notebook style for easy self-study and quick understanding. Each solution is broken down step by step so even the toughest question feels easy. These questions are hand-picked for students preparing for SSC CGL, SSC CHSL, CAT, Bank PO, Bank Clerk, UPSC CSAT, Railway RRB, AMCAT, eLitmus, TCS NQT and all campus placement aptitude tests. International students preparing for GRE, GMAT, SAT, ACT, MAT and all Numerical Reasoning Tests will find these equally useful.

✏️ Attempt each question on your own first — then check the solution below.

11. Find the first non zero digit (FNZD) in 100!

12.Find the first two non zero digit (FNZD) in 100!

13. Find the first non zero digit (FNZD) in 2000!

14. Find the first two non-zero digit(FTNZD) in 2000!

15. Find the first non-zero digit (FNZD) in 20! + 30! +40! + 50! + 60!

16. Find the first non-zero digit (FNZD) in 20! × 30! × 40! × 50! × 60!

17. If the highest power of 10 in N! is 18 then what could be the highest power of 10 in (N+1)!

18. If the highest power of 8 in N! is 19 then find highest power of 8 in (N+1)!

19. A three digit number xyz is such that x + y + z = 26. Then find the sum of all possible values of x!.y!.z!.

20. Find the first two non-zero digit (FTNZD) of \(\frac{{555!}}{{444!}}\)
(i). 12
(ii). 72
(iii). 96
(iv). 38

11. Find the first non-zero digit (FNZD) in 100!.
Solution:-

Hence FNZD(100!) = U𝒹(2²⁴) × U𝒹(0!) × U𝒹(0!) × U𝒹4!)
                                 = 6 × 24
                                 = 4 Answer

12. Find the first two non-zero digit in 100!.
Solution:-

FTNZD(100!) = ?
FTNZD(100!) = Uₜ(12²⁴) × Uₜ(4×3×2×1)
= 2⁴⁸ × 3²⁴ × 24

13. Find the first non-zero digit (FNZD) in 2000!.
Solution:-

FNZD(2000!) = 2⁴⁹⁹ × 0! × 0! × 0! × 1! × 3!
= 8 × 6
= 8           Answer

14. find the first two non zero digit (FTNZD) of 2000!.
Solution:-  FTNZD(2000!) = ?

FTNZD(2000!) = Uₜ(12⁴⁹⁹) × Uₜ(Q₄ × Q₅)

= 76 × 44 × 21 × 27 × 96
= 08       Answer

15. Find first non-zero digit (FNZD) in 20! + 30! + 40! + 50! + 60!.
Solution:-

Hence FNZD of the least factorial number i.e. 20!, will also be the FNZD of the summation.

 ∴ FNZD(20!) = 

= 2⁴ × 4!
= 16 × 24
= 04 Answer

16. Find the FNZD in 20!×30!×40!×50!×60!
Solution:-
In this case the number of trailing zeros in the product will be the summation of number of trailing zeros in individual terms. & FNZD of the product will be the multiplication of the FNZD’s of all the factorial terms.
Hence

∴ FNZD of product = 4×8×2×2×6
= 8         Answer

17. If the highest power of 10 in N! is 18 then what could be the highest power of 10 in (N + 1)!

Solution:-
10 = 2 × 5

Hence number can be N = 75, 76, 77, 78, 79
Hence for (N+1) take N as 79
∴ N+1 = 80

18. If highest power of 8 in N! is 19. Then find the highest power of 8 in (N+1)!.
Solution:-

Highest power of 8 in N! = 19
∴ highest power of 2 in N! = 57, 58 or 59
now we are to find such value of N
To find such N ⇒ 

19. A three digit number xyz is such that x + y + z = 26. Then find the sum of all possible values of x!.y!.z!.
Solution:-

x + y + z = 26
9    9     8   ⇾ 998,       989,       899
∴ there will be three such numbers.
9!.9!.8! + 9!.8!.9! + 8!.9!.9!
= 3.9!.9!.8!      Answer

20. Find the first two non-zero digit (FTNZD) of \(\boldsymbol{\frac{{555!}}{{444!}}}\) .
(i) 12
(ii) 72
(iii) 96
(iv) 38
Solution:- 

FTNZD(555!) =

FTNZD(555!) = Uₜ(12ᴴ) × Uₜ(Q₁×Q₂×Q₃×Q₄)
                          =Uₜ(12¹³⁷) × Uₜ(11×62×24)

∴ FTNZD(555!) = 56

now to find FTNZD(444!) = 

∴ FTNZD(444!) = Uₜ(12ᴴ) × Uₜ(Q₁ × Q₂ × Q₃ × Q₄)

∴ FTNZD(444!) = 48
now to find FTNZD \(\left( {\frac{{555!}}{{444!}}} \right)\)
Let

i.e. 444! × Quotient = 555!
So FTNZD(444!) × FTNZD(Quotient) = FTNZD(555!)
because we know that irrespective of the size of the number & ignoring its all trailing zeros, To find the FTNZD we only need the FTNZD of the numbers & remaining digits of the numbers are useless when finding FTNZD.
∴ 48 × x = 56
now from the given options
(i) 12
(ii) 72 ✔
(iii) 96
(iv) 38

Uₜ(48×72) = 56
Hence option (ii) is correct
∴ FTNZ \(\boldsymbol{\left( {\frac{{555!}}{{444!}}} \right)}\) = 72       option(ii)         Answer

✅ Well done on completing Set 2!

Continue practising with Counting Concept Questions 21 to 30 → Set 3 or revisit the Counting Concept Page to strengthen your formulas and tricks before moving ahead.

Consistent practice is the key to mastering counting concept for SSC CGL, SSC CHSL, CAT, Bank PO, Bank Clerk, UPSC CSAT, Railway RRB, AMCAT, eLitmus, TCS NQT and international exams including GRE, GMAT, SAT, ACT, MAT and all Numerical Reasoning Tests. Want to understand the concept better? Read about Factorial on Wikipedia before attempting the next set.

This page is part of our complete series of counting concept questions with solutions for competitive exams — covering every question type from basic to advanced so you can build speed, accuracy and confidence. Practising these questions regularly will also strengthen your core counting concept before your exam day.