1. No category

Home Work 7 (Discrete)
1- How many binary strings of length 5 contain at least 3 ones?
-
3 ones = C(5, 3)
4 ones = C(5, 4)
5 ones = C(5, 5)
5
5
5
5
5
∑ ( ) = ( ) + ( ) + ( ) = 10 + 5 + 1 = 16
𝑛
3
4
5
𝑛=3
2- How many integer numbers between 100 and 1000 have digits in decreasing order?
All integer numbers from 100 to 1000 consist of three digits, so we are looking for
three digit numbers have their digits in decreasing order such as 321.
(10
) = 120
3
10∗9∗8
OR
=
3!
720
6
= 120
3- Find the number of integer solutions of equation x1 + x2 + x3 = 12 where
𝑥1 ≥ 2, 𝑥2 ≥ 3, 𝑥3 ≥ 4
Replace variables of the equation:
x1 − 2 = 𝑥́ 1
x2 − 3 = 𝑥́ 2
x3 − 4 = 𝑥́ 3
Then find the number of nonnegative solutions of equation
𝑥́ 1 + 𝑥́ 2 + 𝑥́ 3 = 3
(
𝑤ℎ𝑒𝑟𝑒 𝑥́ 1 ≥ 0 𝑥́ 2 ≥ 0 𝑥́ 3 ≥ 0
3+3−1
5
) = ( ) = 10 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑠
3−1
2
4- For given expression (𝑥 + 𝑦)13
a. Find the number of elements of expansion
𝑛+𝑚−1
(
) 𝑤ℎ𝑒𝑟𝑒 𝑛 = 13 (𝑡ℎ𝑒 𝑝𝑜𝑤𝑒𝑟) 𝑎𝑛𝑑 𝑚 = 2 (𝑥&𝑦)
𝑚−1
(
13 + 2 − 1
14
) = ( ) = 14
2−1
1
b. Find the coefficient of element 𝑥 5 𝑦 8
(
13
) = 1287
8
5- How many words with 8 letters can be created from letters F,G,H,I,J
a) That contains substring FG, GH or both?
|𝐴 ∪ 𝐵| = |𝐴| + |𝐵| − |𝐴 ∩ 𝐵|
7
7
7
( ) 56 + ( ) 56 − (( ) ∗ 54 ) =
1
1
2
7 ∗ 15625 + 7 ∗ 15625 − 21 ∗ 625 = 109375 + 109375 − 13125 = 205625
b) Where H appears before I?
(82)
2
∗ 56 =
28
∗ 15625 = 218750
2
6- How many strings of 8 ternary digits (0, 1, or 2) is there that contain exactly two 0s,
three 1s, and three 2s?
C(8,2) * C(6,3) * C(3,3) = 560
7- Find the number of binary strings of length 5 that contain substring ‘011’ or ‘110’.
|𝐴 ∪ 𝐵| = |𝐴| + |𝐵| − |𝐴 ∩ 𝐵|
(31)22 + (31)22 − 0 = 12 + 12 – 0 = 24
8- Find the number of binary strings of the length 8 that contain 5 zeroes, 3 ones, and do
not contain substring "11". Give the reason of your solution
Result = (ALL binary strings of the length 8 that contain 5 zeroes 3 ones) – (Binary
strings of the length 8 that contain 5 zeroes, 3 ones, and contain substring "11")
C(8,5) * C(3,3) – C(7,1) * C(6,1) * 1 =
56 * 1 – 7 * 6 * 1 = 56 – 42 = 14