TheCATExam
Sign in

CAT 2022 Slot 3 — QA Question 8

Numbers (P&C)Easy

The arithmetic mean of all the distinct numbers that can be obtained by rearranging the digits in 1421, including itself, is

Answer & solution

  • 2222

  • B

    2592

  • C

    2442

  • D

    3333

Solution

Easy

The digits of 14211421 are {1,1,2,4}\{1,1,2,4\}. There's a slick shortcut: by symmetry every digit appears in each place equally often, so the average is the average digit repeated — but let's confirm by summing all rearrangements place-by-place.

1

Count the distinct numbers. Four digits with the "11" repeated twice:

4!2!=12 distinct numbers\frac{4!}{2!}=12\text{ distinct numbers}
2

Sum the units digits across all 1212 numbers. Count how often each digit sits in the units place:

1 in units: 3!=6 ways⇒6×1=62 in units: 3!2!=3 ways⇒3×2=64 in units: 3!2!=3 ways⇒3×4=12sum of units digits=6+6+12=24\begin{aligned} &1\text{ in units: }3!=6\text{ ways}\Rightarrow 6\times1=6\\ &2\text{ in units: }\tfrac{3!}{2!}=3\text{ ways}\Rightarrow 3\times2=6\\ &4\text{ in units: }\tfrac{3!}{2!}=3\text{ ways}\Rightarrow 3\times4=12\\ &\text{sum of units digits}=6+6+12=24 \end{aligned}
3

By symmetry each place gives the same 2424, scaled by place value:

Total=24(1+10+100+1000)=24×1111=26664\text{Total}=24(1+10+100+1000)=24\times1111=26664
4

Divide by the count:

Average=2666412=2222\text{Average}=\frac{26664}{12}=2222
Arithmetic mean=2222\text{Arithmetic mean}=\mathbf{2222}

Average digit =1+1+2+44=2=\dfrac{1+1+2+4}{4}=2, so the average number is just 22 in every place: 2222\mathbf{2222}. The repeated digit doesn't change the mean.

CAT 2022 Slot 3 QA Q8: The arithmetic mean of all the distinct numbers that can be obtained by rearranging the digits in 1421, includ — Solution | TheCATExam