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r - Sum of two Columns of Data Frame with NA Values

I have a data frame with some NA values. I need the sum of two of the columns. If a value is NA, I need to treat it as zero.

a  b c d
1  2 3 4
5 NA 7 8

Column e should be the sum of b and c:

e
5
7

I have tried a lot of things, and done two dozen searches with no luck. It seems like a simple problem. Any help would be appreciated!

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dat$e <- rowSums(dat[,c("b", "c")], na.rm=TRUE)
dat
#   a  b c d e
# 1 1  2 3 4 5
# 2 5 NA 7 8 7

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