How many strings are there of four decimal digits that:

a) Exactly contains four 9-s.

b) Exactly 2 digit must be same

c) The string ends with a even digit greater than 2

My solutions:

a) $$\binom9 4= \frac{9!}{4!(9-4)!}= 126$$

b) First digit - 10 ; Second digit - 10 ; Third digit - 9 ; Fourth digit - 8 --- So: $$10*10*9*8= 7200$$

c) First digit - 7 ; Second digit - 6 ; Third digit - 5 ; Fourth digit - 4 --- So: $$7*6*5*4 = 840$$

Please correct me if I'm wrong!!!

a)If a 4 digit number contains exactly four 9's , then $$^9C_4$$ is not your solution , there is only one way i.e 9999 , so answer should be 1
b)This one is not as simple as you did,exactly two same will have conditions either $$1^{st}$$ and $$2^{nd}$$ are same or $$1^{st}$$ and $$3^{rd}$$ are same and so on , there will be 6 such conditions , you will have to solve them separately , .
c)It's nowhere mentioned that digits other than last one cannot be even so answer should be 10 x 10 x 10 x 3 , for (4,6,8 viz. even digits greater than 2)

• The title refers to strings of four digits, not numbers containing four digits. The first digit can be zero in a string. Feb 3 '19 at 10:06
• When I had answered the question , it was numbers and not decimals , the question was edited i guess. Feb 3 '19 at 10:28

a) There is only one string of length $$4$$ that contains four nines, "9999", so the answer is $$1$$.

b) If exactly two digits are the same.: first choose the two places out of $$4$$ that are going to contain the same digit in $$\binom{4}{2}=6$$ ways.

Then pick the digit that goes in those $$2$$ places : $$10$$ choices, then we have $$9$$ and $$8$$ choices for the two remaining positions (in left to right order, for definiteness). So finally we have $$6 \times 10 \times 9\times 8$$ ways to choose, which gives us $$4320$$ strings.

c) All first three digits are free to choose, independently, the last one is one of $$\{4,6,8\}$$ so has only $$3$$ options. So $$3 \times 10^3 = 3000$$ strings.

• @N.F.Taussig you could have edited it too, I don’t mind. Thx. Feb 3 '19 at 10:31
• Thank you so much.. Feb 3 '19 at 10:32