Count pairs occurence and plot them

I have this dataset called x for example:

I then want to create another table, that calculates the number of occurence of each possible pair in the table. So the result would be something like this:

That means for example that for the first row, a occured (was equal to 1) alone (means other columns didn't, and so they were at 0) only once.

After this, I want the best way to visualize the result of the pairs in ggplot.

Edit: More explanation of the second table:

a (single letter): a is the strictly one which occured (rest are 0).
a,b (two letters): a and b both occured, but not the rest.
a,b,c (three letters): all a,b,c occured at the same time (all are set to 1).

Here's the pivoting part, not sure of the best way (atm) to plot.

library(dplyr)
poss <- data.frame(pair=unlist(sapply(seq_along(names(quux)[-1]), function(z) combn(x = names(quux)[-1], z, FUN = function(y) paste(y, collapse=",")))))
poss
#    pair
# 1     a
# 2     b
# 3     c
# 4   a,b
# 5   a,c
# 6   b,c
# 7 a,b,c


library(dplyr)
library(tidyr)
quux %>%
  pivot_longer(-client_id) %>%
  summarize(pair = paste(sort(unique(name[value==1])), collapse = ","), .by = client_id) %>%
  count(pair) %>%
  full_join(poss, by="pair") %>%
  mutate(n = coalesce(n, 0)) %>%
  arrange(nchar(pair), pair)
# # A tibble: 7 × 2
#   pair      n
#   <chr> <dbl>
# 1 a         1
# 2 b         0
# 3 c         1
# 4 a,b       0
# 5 a,c       0
# 6 b,c       1
# 7 a,b,c     1

Data

quux <- structure(list(client_id = 1:4, a = c(1L, 0L, 0L, 1L), b = c(0L, 1L, 0L, 1L), c = c(0L, 1L, 1L, 1L)), class = "data.frame", row.names = c(NA, -4L))