XPath/XSLT contains() for multiple strings

Assuming that you're looking to do a logical OR of substring containment...

First of all, this select is likely wrong¹:

<xsl:variable name="games" select="footballvolleyballchesscarrom"/>

It's looking for an element named footballvolleyballchesscarrom. If you're trying to set games to the string footballvolleyballchesscarrom, change it to this:

<xsl:variable name="games" select="'footballvolleyballchesscarrom'"/>

Then, use either

XSLT 1.0

<xsl:when test="contains($games,'chess')
             or contains($games,'carrom')">CORRECT</xsl:when>

or

XSLT 2.0

<xsl:when test="matches($games,'chess|carrom')">CORRECT</xsl:when>

to test whether $games contains one of your multiple substrings.

¹ If you really intended to select (possibly multiple) elements from the input XML here, there is another XSLT 2.0 option for testing the selected sequence against a set of possible values. If your input XML were, say,

<games>
  <game>football</game>
  <game>volleyball</game>
  <game>chess</game>
  <game>carrom</game>
</games>

then this template,

<xsl:template match="/">
  <xsl:variable name="games" select="//game"/>
  <xsl:if test="$games = ('chess','carrom')">CORRECT</xsl:if>
</xsl:template>

would output CORRECT because among the string values of the selected elements is at least one of chess or carrom.

I think multiple variables will not work with contains method. As per the defination:

boolean contains(str1, str2)

str1 : A string that might contain the second argument. str2: A string that might be contained in the first argument.

Refer : https://msdn.microsoft.com/en-us/library/ms256195(v=vs.110).aspx

You can use or operator:

<xsl:when test="contains($games,'chess') or contains($games,'carrom')">CORRECT</xsl:when>