Sum of a nCk sequqnce with changing n

How to proof the following without using induction?

This is expected value of a probability question, I found the R.H.S just by looking out the results when m=1,2,3,4,…..
But I cannot find any way to simplify LHS to RHS

For all m, n >= 1
Big{sum_{i=0}^{n-1} {m-1+i choose m-1} (n-i) Big} frac{1}{m+n-1 choose m} = frac{n+m}{m+1}

For example, when m=4,n=3,

L.H.S. &= big{{3 choose 3}(3) + {4 choose 3}(2) + {5 choose 3}(1)big}frac{1}{{6 choose 4}}\
&={(3)(3) + (4)(2) + (10)(1)}/15\
&= 7/5\
R.H.S. &= (4+3)/(4+1) = 7/5

windows 10 – How to stop PowerPoint from being “smart” and constantly changing proofing language?

I know this may sound like a duplicate of Stop Powerpoint from changing proofing language to keyboard language but the solution there doesn’t seem to work.
I have Windows 10 20H2, build 9042.867 and PowerPoint v.2103.
My system language is English, and I have Russian installed as well.
My presentation is in Russian, but when I try changing the proofing language to Russian, it changes back to English within like a second. I have moved the auto detection to th bottom of the Spellcheck list in options, and Russian to the top. It doesn’t help. I have tried changing keyboard layout, but that didn’t work either.
This presentation is for a class, so grammar is very important to me. Please help!

default settings – Changing the Microsoft Word past option for Excel tables?

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seo – Search Console: What to do when changing from subdomain to TLD

I found this post but I it doesn’t answer my question completely.
I have

  • and others

which I moved using 301 to


The redirects work perfectly.

Currently do exist in the Google Search Console the following sitemaps URL:


In the Search Console I already added/confirmed,, … as new properties.

My questions:

  1. Should I keep the existing sitemaps for the subdomains [] or delete them?
  2. Is there a way to tell the Search Console that is the new or how to do it?
  3. It’s obvious I don’t want to lose the ranking for the subdomains and inherit them to the new TLDs.
  4. What do I have to consider, what do I have to do?
  5. Can I already submit the sitemap or should I wait?

Will changing a site from no-www to using www affect SEO?

My client wants to switch their site from to https:/// I know how to make the switch to having www in front of the domain – but does anyone know if making this switch will affect Google Rankings?

settings – How can I prevent my soft keyboard from changing for different input types?

I would like to disable the soft keyboard mode (gboard) for inputType=textUri on my Pixel 4a, and fall back on the default text keyboard mode.

Coming from a much older version of Android which supported fewer inputTypes, I am finding it difficult to adjust to these similar-yet-subtly-different soft keyboards. In particular, the uri mode is causing problems for me, as mobile browsers tend to use it in their top url/search bar. This causes me problems because punctuation commonly used for logic in search, such as “, +, -, :, !, etc, are removed from their usual location (on my keyboard, behind the period), and replaced with shortcuts for common top level domains.

I would like my keyboard to always be the same, or at least, I would like to disable some of the input modes that are similar enough to be confusing.

I understand that this input type feature may not be intended to be configurable.
Still, I would appreciate any advice that can be offered.

long exposure – Issues with dark frame subtraction: Dark frames adding “noise” and changing image color/tint

While editing some landscape shots with stars, I tried to use darkframes to reduce the noise.
More precisely, my approach was to take a series of shots, then firstly to subtract dark frames from each shot, secondly to use the mean of the series for the foreground to further reduce noise, and thirdly to use an astro stacking tool (Sequator) to stack the sky.

Instead of reducing noise, the darkframe subtraction:

  1. increased the noise- or rather, added some dark/monochrome noise.
  2. changed the white-balance/tinted the image.
    (see below)
    I do not understand why this is happening/What I am doing wrong.

Procedure/Employed Troubleshooting:

  • All photos were shot in succession, with the same settings (15sec, @ISO6400, in-camera dark frame disabled).

  • All photos were shot with the same white balance.

  • While shooting the darkframes, both the lens cap and the viewfinder cover were applied.

  • Photos were imported from my Pentax K1ii, converted to DNG in LR, and exported to PS without any editing/import presets applied.

  • I used PS, placed the darkframe layer(s) above my picture, and used the subtract blending mode.

  • I followed basic instructions found here/in various videos on dark frame subtraction in photoshop. Note that basically, all of those cover dark frame subtraction with one frame (or use tools other than photoshop). I have tried both using one, and 3 frames. The results are similar, albeit more pronounced with 3.

  • I used the free tool “sequator” to subtract darkframes instead (and to align stars). Adding the dark frames here made absolutely no difference.

  • (This is an edit/composite done with the frames I tried to subtract darkframes of)

  • A crop of the first picture, with (3) dark frames subtracted:
    with (3) dark frames

  • A crop of the second picture, without dark frames subtracted:
    without dark frames

How to reload the shortcode content when the database is changing

I use WordPress plugin: snippets to show some data in database. Everything is OK, however, when I change the content in the database. After that I refresh many time in the page. And the content is not change. I close the web page, and wait for 5 minutes. Reopen the page again, the content is updated.
What happens in my situation? Please give me some suggestion!

In Snippets:

public function some_function() {
    global $wpdb;
    $results = $wpdb->get_var( 
        $wpdb->prepare("SELECT count(ID) as total FROM {$wpdb->prefix}sample_table") 
    return $results;
add_shortcode('sample_shortcode', 'some_function');

In the Page – I add the shortcode


dnd 5e – Changing griffon feeding habits

dnd 5e – Changing griffon feeding habits – Role-playing Games Stack Exchange

dnd 5e – Does “Changing a skill” of Tasha allows me to changing a skill proficiency from the background for one offered by the class?

It’s up to the DM, but the language does not contain an explicit restriction.

This section is actually quite short:

Sometimes you pick a skill proficiency that ends up not being very useful in the campaign or that no longer fits your character’s story. In those cases, talk to your DM about replacing that skill proficiency with another skill proficiency offered by your class at 1st level.

That this is entirely up to the DM is hard coded into this rule. That said, the language does not contain any restriction on which skills may be replaced, it simply says “sometimes you pick a skill”.

So this isn’t like an ability score improvement where I just pick what I want. I don’t have to talk to my DM when making a decision about an ability score improvement. Working with your DM on this is just part of the process.

Personally, I see no reason whatsoever to not allow you to change any skill proficiency you want. But your DM may see a reason, and that is okay too.

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