Answered: How many bit strings of length 10…

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Question

How many bit strings of length 10 contain
a) exactly four 1s?
b) at most four 1s?
c) at least four 1s?

Expert Solution

Step 1

a) Exactly four 1s

This just asks us to choose four out of ten places in which to place one.

O(10, 4) = 10!/(4! × 6!)

= (10 × 9 × 8 × 7)/4! = 210

b) At most four 1s

We count the number of bit strings with zero 1s, one 1, two 1s, three 1s, and four 1s in length.

O(10, 0) + C(10, 1) + C(10, 2) + C(10, 3) + C(10, 4)

= 10!/(0! × 10!) + 10!/(1! × 9!) + 10!/(2! × 8!) + 10!/(3! × 7!) + 10!/(4! × 6!) =

=1 + 10 + 45 + 120 + 210 = 386.

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