How many can you guess?

Number Theory Level pending

A ternary number is a sequence of digits, each either $2, 1,$ or $0$ . How many $8$ -digit ternary numbers have exactly three $1$ (one)?

Examples of an $8$ -digit ternary numbers are $10102100$ and $12012101$ and $02011100$ .

1 solution

A Former Brilliant Member
Nov 3, 2017

Think of building such a number in two stages.

Stage 1: Specify the location of the three $1$ . There are $_8C_3=56$ ways to do it.

Stage 2: Specify the remaining digits. There are $2^5=32$ ways to do it.

By the product rule, there are $56 \times 32=1792$ eight-digit ternary numbers that have exactly three $1$ .