Q:

Discrete Math Strings of length 10 (Logical Question)

Original Question:
For the strings of length 10 in how many
have:

a) four O's, three 1 's, and three 2'?
b) at least eight l's ?
c) weight 4 ?

Textbook Solution:
a) 10!/4!3!3! (arrangements with repeated symbols)
b)$10\choose8$$2^2$ + $10\choose9$$2^1$ + $10\choose10$$2^0$
c)$10\choose4$(four 1's, six 0's) + $10\choose2$$8\choose1$ (two 1's, one 2, seven 0's) + $10\choose2$(two 2's, eight 0's)
Textbook's Background Information: (Hint)

Assume the string is the length 10, and it is made of 0's, 1's, and 2's.
"In general, if n is any positive integer, then by the rule of product there are $3^n$ strings of length n for the alphabet 0, 1, 2".
"If x = x 1 x2 x3 · · · Xn is one of these strings, we define the
weight ofx, denoted wt(x), bywt(x) = x 1 + x2 + x3 + · · · + Xn. For example, wt(l2) = 3
and wt(22) = 4 for the case where n = 2; wt(101) = 2, wt(210) = 3, and wt(222) = 6 for
n = 3".

Personal Logical Question:
I have problem understanding the solution and logic for part C.
My guess would be weight of 4 is referring to four 1's or two 1's and one 2's or two 2's. Am I right ?

A:

The weight is just the sum of the entries in the string.  It appears that the strings are only composed of $0,1,2$, though you don't say that.  In that case, a weight of four can then be achieved by four $1$s, a $2$ and two $1$s, or two $2$s and the rest zeros.  The text is counting the number of ways to positions those in a string of $10$ for each case.

