@words_which_start_with_double_quotes = grep {/^"/} @array_of_words;
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but it is not working , @foo_d always holds 1 if anyone can please help out

GIGO, Basic debugging checklist , brian's Guide to Solving Any Perl Problem

@arr = qw/ ""fa la la ""la /;
@foo_d = grep(/""/, @arr);
print $_,$/ for @foo_d;
__END__
""fa
""la
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You should open your own thread if the question is different (here you just have a common topic). Maybe it's a bit late far that this time, but for your next question maybe :).

So, if you want a string that starts with two double quotes, that would be with /^""/. And then, if you have a number as a result instead of strings, you probably called grep in a scalar context (so you get the number of matches instead of the matches). Maybe you wrote $foo_d instead of @foo_d, or you forgot a ; in the previous line. Or maybe you called @foo_d in a scalar context when printing it, you'd still get a number of elements then instead of content.

Not working because you are looking for two double quotes instead of one double quote. If you want a double quote at the start of your string:

You really need to show your input and output as well as the process you're using to get from one to the other. Guidelines for doing this can be found in "How do I post a question effectively?".

"but it is not working , @foo_d always holds 1 if anyone can please help out"

How are you determining what "@foo_d always holds"? Here's a number of different ways that you might be doing this:

$ perl -Mstrict -Mwarnings -le '
    my @all_words = qw{abc "def ghi "jkl mno};

    print "*** Using grep BLOCK LIST ***";
    my @qq_words_block = grep { /^"/ } @all_words;
    print "qq_words_block = ", @qq_words_block;
    print "qq_words_block = " . @qq_words_block;
    print "qq_words_block = ", "@qq_words_block";
    print "qq_words_block = " . "@qq_words_block";

    print "*** Using grep EXPR, LIST ***";
    my @qq_words_expr = grep(/^"/, @all_words);
    print "qq_words_expr = ", @qq_words_expr;
    print "qq_words_expr = " . @qq_words_expr;
    print "qq_words_expr = ", "@qq_words_expr";
    print "qq_words_expr = " . "@qq_words_expr";
'
*** Using grep BLOCK LIST ***
qq_words_block = "def"jkl
qq_words_block = 2
qq_words_block = "def "jkl
qq_words_block = "def "jkl
*** Using grep EXPR, LIST ***
qq_words_expr = "def"jkl
qq_words_expr = 2
qq_words_expr = "def "jkl
qq_words_expr = "def "jkl
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In list context, @array expands to its elements; in scalar context, you get the number of elements.

Search for $LIST_SEPARATOR in perlvar for a discussion of why "@array" separates the elements with spaces.